The Physics of Guitar Tone

For a string of sounding length L displaced to height h at relative position R = a/L and released, initial modal amplitude is C_n ∝ (2h/n²π²R(1−R))·sin(nπR): 1/n² is the corner's roll-off, 1/[R(1−R)] a scalar gain, sin(nπR) the comb that zeroes every m-th harmonic when L/a = m; the first spectral maximum sits at f0/2R with later maxima at odd multiples. Measured on an electric guitar's magnetic pickup (13th harmonic absent at a 14.8 cm pluck on a 64.3 cm string) and on an acoustic string across 21 positions × 5 trials (harmonic amplitude tracks standing-wave amplitude at the pluck point). Bounds: the exciter has width so notches never reach zero; bridge nonlinearity regenerates missing modes over about 1 s so the comb is an attack-window property; the pickup is a second spatial sampler at x_pu whose factor sin(nπx_pu/L) multiplies the hand's comb (derivation). Because the string is transversely stiffer near an end (F ≈ Th(1/a + 1/(L−a))), 'same force' and 'same displacement' plucks are different experiments and 'near the bridge is brighter' is a normalized-spectrum statement, not a level statement. Population: physical measurements on single instruments (one electric, one acoustic, one robot); outcome: measured harmonic amplitudes; class: MODEL confirmed by INTERVENTION.

A triangular initial shape has a spatial Fourier series whose coefficients vanish for modes with a node at the kink; finite contact width convolves the kink and fills the null; the quadratic nonlinearity at a moving termination couples modes and refills them; a magnetic pickup senses string velocity at one point, so its own sin(nπx_pu/L) factor multiplies the launched spectrum.

• S1Woodhouse, Euphonics: Guitar I (plucked-string model, two polarizations, endpoint mobility; site 2022)MODEL: the stiff damped string equation, the triangular-pluck modal law, the two-polarization pluck-direction synthesis and the near-end transverse stiffness F ≈ Th(1/a + 1/(L−a)).
• S5Carral, Plucking the String: The Excitation Mechanism of the Guitar, ASA 160th meeting lay-language paper, 2010 (reporting Carral and Paset, JASA 123(5), 2008)INTERVENTION: machine plucks with 0.46 mm and 2 mm plectra; amplitude and spectral centroid differed and the centroid ordering reversed after about one second; gradual release favors the fundamental, sudden release excites high modes.
• S13Traube and D'Alessandro, Vocal synthesis and graphical representation of the phonetic gestures underlying guitar timbre description, DAFx-05, 2005MODEL: the ideal plucked-string spectral envelope is comb-shaped with first maximum F_C1 = f0/2R and later maxima at odd multiples.
• S16Fletcher, The nonlinear physics of musical instruments, Rep. Prog. Phys. 62:723 to 764, 1999 (text opened 2026-10-05)MODEL + cited measurement: modes missing at excitation are regenerated by nonlinear effects at the bridge over a time of order 1 s (citing Legge and Fletcher, JASA 1984).
• S18Aldea, Plucking Location and Timbre of a Guitar, International Scholastic Journal of Science 15(1), 2021 (text opened 2026-10-05)INTERVENTION: an acoustic guitar E4 string plucked at 21 positions, 5 trials each, microphone at 100,000 samples/s; harmonic amplitudes follow the standing-wave amplitude at the pluck point and never reach zero because the exciter has width.
• S19Perov, Johnson, Perova-Mello, The physics of guitar string vibrations, American Journal of Physics 84:38, 2016 (Crossref-verified; opened by the anthropic leg via ResearchGate, publisher page 403 for this writer)INTERVENTION: with L = 64.3 cm and a pluck at 14.8 cm the 13th harmonic was absent from the magnetic pickup's FFT, the predicted sin(nπa/L) = 0 null (as quoted by the anthropic leg).
• S21Wiciak, Małecki, Tokarczyk, Impact of a guitar string pluck on the instrument's tone, Tecniacústica Oporto, 2016INTERVENTION: a guitar-playing robot with a movable plucking module varied pluck position and pick angle because human repeatability was too low; position changed the spectral bands observed.

Pluta, Jasiński, Tokarczyk and Grygiel (Vibrations in Physical Systems 36(2), 2025; arXiv 2606.24356, 2026-06-23) moved a robotic plucker toward an acoustic guitar in 192 μm attack-depth steps with plectra of six materials and measured loudness, timbre descriptors, harmonic content and decay. At low depth the string is not fully excited and the sound is weak and markedly altered; the width of that band depends on plectrum material; past it each depth step raises loudness, lowers inharmonicity and noisiness and moves timbre toward fuller low frequencies and more roughness while decay shape stays similar. The transfer from gesture to sound is therefore gated and qualitatively different below threshold. Boundary: one acoustic guitar, one robot, a microphone chain; transfer to a solidbody with a magnetic pickup is UNVERIFIED; the authors warn remounting plectra broke cross-pick comparison and the arXiv referee note leaves open whether depth was isolated from speed or angle. Outcome: measured loudness, inharmonicity, noisiness, timbre descriptors; class: INTERVENTION (controlled physical measurement) with stated confounds.

A plectrum must deflect the string past the point where it slips off; below that depth the release is partial and the initial displacement and velocity distributions are small and irregular; above it, larger displacement raises mean tension (B ∝ 1/T), lowering measured inharmonicity, and increases low-mode energy.

• S3Bank, Energy-based synthesis of tension modulation in strings, DAFx-09, 2009MODEL: ΔT ≈ (EA/2L)∫|u_x|² tension modulation, the geometric nonlinearity behind the onset pitch glide.
• S10Pluta, Jasiński, Tokarczyk, Grygiel, The effect of micro-changes in the pluck trajectory on the sound of an acoustic guitar, Vibrations in Physical Systems 36(2):2025205, 2025INTERVENTION: robotic plucks at 192 μm attack-depth steps with six plectrum materials on an acoustic guitar; below a material-dependent depth the string is incompletely excited; above it loudness rises, inharmonicity and noisiness fall, timbre fills in lows and roughens; remounting plectra broke cross-pick comparison.
• S24Tolonen, Välimäki, Karjalainen, Modeling of tension modulation nonlinearity in plucked strings, IEEE Trans. Speech and Audio Processing, 2000-05 (DOI 10.1109/89.841212)MODEL: tension modulation produces an exponentially decaying pitch glide, longitudinal modes and phantom partials; the glide is an important timbral feature of the electric guitar among others.
• S25Pluta, Jasiński, Tokarczyk, Grygiel, arXiv 2606.24356, submitted 2026-06-23 (arXiv version of S10)INTERVENTION (same study as S10); the public referee note leaves open whether depth was isolated from pick speed or angle.

A real pluck leaves the string with both a displacement and a velocity distribution at release, and those, not peak force, determine the free motion. Carral's machine plucks (ASA 2010, reporting Carral and Paset JASA 2008) with 0.46 mm and 2 mm plectra: the thin plectrum bent more, the thick one displaced the string farther, amplitude and spectral centroid differed and the centroid ordering reversed after about one second; release timing depends on plectrum elasticity and friction so the same depth gives different initial amplitude and modes present; gradual release enhances the fundamental, sudden release excites higher modes. 'Pick angle' must be operationalized as edge orientation, travel direction, and the string's displacement and velocity direction at release, which are not interchangeable. The AGH robot (2016) varied angle and position repeatably. The anthropic leg quotes an acoustic-guitar angle sweep (little change until a steep fall near the parallel limit; perpendicular plucks strong and fast-decaying, parallel weaker and longer) that this writer could not pin to an opened URL, so its angle and decibel figures carry no number here and the envelope half rests on A4. Cuzzucoli and Lombardo's model parameters (plucking force 1 to 5 N, exciter mass 0.5 to 3 g, finger stiffness up to about 3,000 N/m, displacement about 1 to 5 mm) are measured on one classical guitar, not population values. Class: INTERVENTION (machine plucks) + MODEL (contact release).

The string's free motion is set by its state at the instant contact ends; a release that unloads over time τ suppresses energy above about 1/τ, while a sudden slip or recontact injects a broadband transient; a stiff, perpendicular plectrum minimizes τ and a flexible, slanted one maximizes it.

• S1Woodhouse, Euphonics: Guitar I (plucked-string model, two polarizations, endpoint mobility; site 2022)MODEL: the stiff damped string equation, the triangular-pluck modal law, the two-polarization pluck-direction synthesis and the near-end transverse stiffness F ≈ Th(1/a + 1/(L−a)).
• S5Carral, Plucking the String: The Excitation Mechanism of the Guitar, ASA 160th meeting lay-language paper, 2010 (reporting Carral and Paset, JASA 123(5), 2008)INTERVENTION: machine plucks with 0.46 mm and 2 mm plectra; amplitude and spectral centroid differed and the centroid ordering reversed after about one second; gradual release favors the fundamental, sudden release excites high modes.
• S6Evangelista and Eckerholm, Player-Instrument Interaction Models for Digital Waveguide Synthesis of Guitar Touch and Collisions, IEEE TASLP, 2009MODEL: the finger or plectrum as a damped spring-mass system with time-varying parameters, fret collisions and imperfect stopping.
• S21Wiciak, Małecki, Tokarczyk, Impact of a guitar string pluck on the instrument's tone, Tecniacústica Oporto, 2016INTERVENTION: a guitar-playing robot with a movable plucking module varied pluck position and pick angle because human repeatability was too low; position changed the spectral bands observed.
• S23Cuzzucoli and Lombardo, Physical model of the plucking process in the classical guitar (ICMC 1997; CMJ 1999)MODEL with empirically measured parameters: plucking force 1 to 5 N, exciter mass 0.5 to 3 g, finger stiffness up to about 3,000 N/m, displacement about 1 mm (piano) to 5 mm (forte) on one classical guitar.

Every string mode is a pair, one polarization normal to the body and one parallel, at slightly different frequencies with different damping because the termination's admittance differs by direction; the player sets the initial mixture by the pluck's plane. Where vertical bridge admittance exceeds horizontal, energy leaves the vertical polarization faster and the sum shows a fast onset decay then a long sustain, with beating where the pair's frequencies differ. Woodhouse's necessary condition (JASA 150(6):4375, 2021): both polarizations share the same intrinsic damping, so a strong contrast in total damping requires one to lose energy through the bridge more than the other by at least the intrinsic damping. Pick plane therefore controls the decay envelope if and only if the termination is that anisotropic; on an isotropic high-impedance termination the same gesture changes spectrum and leaves the envelope unchanged. Measured: Paté, Le Carrou and Fabre (2014) resolved both polarizations of electric-guitar strings on a rigid fixture with two optical sensors and separated intrinsic loss from structural coupling; an Acta Acustica 2020 rig measured polarization coupling through a half-round saddle with two laser Doppler vibrometers; Le Carrou's group (2019) measured that the player's grasp, strap and thigh contact change the electric guitar's modal damping and frequencies. Intrinsic damping varies by an order of magnitude across the audible range. The solidbody loss partition (intrinsic, contact, structure) is UNVERIFIED. Population: single electric instruments and fixtures; outcome: measured per-polarization decay rates; class: MODEL (theorem) confirmed by INTERVENTION.

A termination with direction-dependent admittance drains energy from the two orthogonal polarizations at different rates; the observed envelope is the sum of two exponentials whose separation equals the admittance contrast minus the shared intrinsic loss, so anisotropy below intrinsic loss yields a single decay whatever the pluck plane.

• S1Woodhouse, Euphonics: Guitar I (plucked-string model, two polarizations, endpoint mobility; site 2022)MODEL: the stiff damped string equation, the triangular-pluck modal law, the two-polarization pluck-direction synthesis and the near-end transverse stiffness F ≈ Th(1/a + 1/(L−a)).
• S2Paté, Le Carrou, Fabre, Predicting the decay time of solid body electric guitar tones, JASA 135(5):3045 to 3055, 2014INTERVENTION: polarization-resolved decay of electric-guitar strings on a rigid fixture with two optical sensors, separating intrinsic string loss from structural coupling.
• S9Le Carrou, Paté, Chomette, Influence of the player on the dynamics of the electric guitar, JASA, published 2019-11-06INTERVENTION: the player's grasp, strap and thigh contact change the electric guitar's modal frequencies and add modal damping, depending on body part and mode shape.
• S14Acta Acustica 2020, article aacus200019 (polarization coupling through a half-round saddle; title not verified by this writer, publisher page 403 on 2026-10-05)INTERVENTION: coupling of the two transverse polarizations through a plain half-round saddle measured with two laser Doppler vibrometers and a reproducible plectrum pluck (as quoted by the anthropic leg).
• S15Woodhouse, A necessary condition for double-decay envelopes in stringed instruments, JASA 150(6):4375, 2021-12-01 (DOI 10.1121/10.0009012, Crossref-verified)MODEL (theorem): a strong contrast in damping between the two polarizations requires one to lose energy through the bridge more than the other by at least the intrinsic damping.
• S17Woodhouse, Euphonics 7.3 Multiple strings and double decaysMODEL: each string mode is a pair of polarizations with different frequencies and damping; unequal bridge admittance gives a two-stage envelope; the player sets the initial mixture by the gesture.
• S26Woodhouse, Euphonics: Guitar IIMODEL + measurement: intrinsic string damping varies by an order of magnitude across the audible range, so no single decay constant describes a gesture's effect.
• S27Smith, Physical Audio Signal Processing: String Coupling Effects (online book)MODEL: vertical and horizontal polarizations couple to the bridge with different admittance, giving fast then slow decay and beating.

With axially immobile ends, small transverse slopes add length Δℓ ≈ ½∫|u_x|² dx and tension ΔT ≈ (EA/2L)∫|u_x|² dx; large-amplitude onset motion raises mean tension and pitch, which glides down as amplitude decays (Bank DAFx-09 2009; Tolonen, Välimäki, Karjalainen IEEE TSAP 2000, who name the glide an important timbral feature of the electric guitar). For a dominant mode with A(t) ∝ e^(−γt) the offset falls as e^(−2γt). For a triangular pluck of height h at distance p from the bridge, Δf/f ≈ (1/2ε0)·h²/[2p(L−p)] with ε0 = T/EA, so the offset is quadratic in depth and larger near the bridge at equal depth: force and position interact on pitch (anthropic-leg derivation). The same geometric nonlinearity produces longitudinal modes, phantom partials and mode coupling; the material stays Hookean. Magnitudes: Legge and Fletcher's nylon result that a 10 mm deflection is reachable only by pulling bounds the effect as small at ordinary classical dynamics; for steel electric strings no opened source measured the glide at a stated depth, so the circulating figures (a blog model's 26 cents at 4.7 mm; a derived 8 to 9 cent upper bound at h = 3 mm, p = L/5, L = 0.635 m, ε0 ≈ 0.007) are MODEL outputs only. Friction can partition tension across nut or saddle, qualifying a uniform-tension model. Class: MODEL; the only measurements are model fits to recorded tones.

Transverse displacement lengthens the string by a second-order amount, which through the axial stiffness EA raises tension in proportion to the squared slope integral; wave speed √(T/μ) rises with it, so pitch tracks the squared amplitude and relaxes twice as fast as amplitude.

• S1Woodhouse, Euphonics: Guitar I (plucked-string model, two polarizations, endpoint mobility; site 2022)MODEL: the stiff damped string equation, the triangular-pluck modal law, the two-polarization pluck-direction synthesis and the near-end transverse stiffness F ≈ Th(1/a + 1/(L−a)).
• S3Bank, Energy-based synthesis of tension modulation in strings, DAFx-09, 2009MODEL: ΔT ≈ (EA/2L)∫|u_x|² tension modulation, the geometric nonlinearity behind the onset pitch glide.
• S4Kemp, The physics of unwound and wound strings on the electric guitar applied to the pitch intervals produced by tremolo/vibrato arm systems, PLOS ONE, 2017-09-21INTERVENTION + MODEL: the string slips over nut and saddles to equalise tension within the limits of friction, leaving imbalanced tensions and tuning problems after a tremolo excursion.
• S16Fletcher, The nonlinear physics of musical instruments, Rep. Prog. Phys. 62:723 to 764, 1999 (text opened 2026-10-05)MODEL + cited measurement: modes missing at excitation are regenerated by nonlinear effects at the bridge over a time of order 1 s (citing Legge and Fletcher, JASA 1984).
• S24Tolonen, Välimäki, Karjalainen, Modeling of tension modulation nonlinearity in plucked strings, IEEE Trans. Speech and Audio Processing, 2000-05 (DOI 10.1109/89.841212)MODEL: tension modulation produces an exponentially decaying pitch glide, longitudinal modes and phantom partials; the glide is an important timbral feature of the electric guitar among others.

Regime A, contact: the force to bring a string to the fret is F ≈ T1 sin θ1 + T2 sin θ2, set by action, finger position, tension and fret geometry; action dominates and higher frets need more force (Grimes 2014). Regime B, hold: a fret is a unilateral contact (g ≥ 0, N ≥ 0, gN = 0), so static touch does not guarantee the boundary during a large pluck; detachment, recontact and collision with the next fret are nonlinear and the threshold depends on excitation amplitude and direction (Evangelista and Eckerholm 2009; Evangelista 2011); the anthropic leg writes F_hold > T(d/a + d/b) + κTh(1/p + 1/(L−p)), so fretting pressure adequate for soft picking chokes under hard picking: the two hands couple through the boundary (derivation). Regime C, excess force: pressing past contact lengthens the path behind the fret and sharpens the note only insofar as the extension raises tension in the sounding segment, with fret friction and anchor compliance governing transmission; the capstan law with a fraction-of-a-degree wrap at a fret equalizes tension almost freely while the nut's larger wrap resists slip, so excess-force sharpening is near-reversible and bend redistribution across the nut is not (derivation). The unavoidable contact stretch that action imposes on every fretted note is a setup constant removed by saddle compensation, not a gesture; the player's excess force is a few cents by geometry for half a millimeter of over-press between supports 8 and 25 mm apart (derivation). Circulating figures (14 to 15 cents from forums; 'a quarter tone' from an instructional site) are ARGUMENT and carry no number. A force-sensing fretboard (Ren 2023) reads 0 to 25 N at 20 Hz on compliant frets and cannot resolve attack mechanics. Finger-applied force is not the fret reaction: skin compression and fingerboard contact carry part of the load. Class: MODEL with S7's bend leg measured; instrumentation CORRELATIONAL.

A fret pushes but cannot pull, so the boundary holds only while the string's transverse force at the fret stays below the finger's normal load; pressing past contact adds path length quadratically in depression and raises tension only where friction lets the segments equalize.

• S4Kemp, The physics of unwound and wound strings on the electric guitar applied to the pitch intervals produced by tremolo/vibrato arm systems, PLOS ONE, 2017-09-21INTERVENTION + MODEL: the string slips over nut and saddles to equalise tension within the limits of friction, leaving imbalanced tensions and tuning problems after a tremolo excursion.
• S6Evangelista and Eckerholm, Player-Instrument Interaction Models for Digital Waveguide Synthesis of Guitar Touch and Collisions, IEEE TASLP, 2009MODEL: the finger or plectrum as a damped spring-mass system with time-varying parameters, fret collisions and imperfect stopping.
• S7Grimes, String Theory: The Physics of String-Bending and Other Electric Guitar Techniques, PLOS ONE 9(7):e102088, 2014-07-23INTERVENTION (bend pitches measured at fixed displacements on an electric guitar) + MODEL (fretting-force equation, action dominance, radial and axial vibrato simulation, whammy-bar tension).
• S8Evangelista, Physical model of the string-fret interaction, DAFx-11, 2011MODEL: the fret clamps the vertical polarization while the horizontal stays free under friction; the finger adds damping and clamps the trailing segment.
• S12Ren et al., force-sensing fretboard, arXiv 2304.06521, 2023CORRELATIONAL (instrumentation): per-fret contact force read 0 to 25 N at 20 Hz on compliant sensing frets; does not preserve an ordinary fret boundary or resolve attack mechanics.

A stationary dashpot c_f at x_f adds to mode n a decay-rate increment Δγ_n = (c_f/μL)·sin²(nπx_f/L): a mode with a node at the contact is barely touched, one with an antinode strongly damped (weak-contact model; the finger as a spring-mass-damper in Evangelista and Eckerholm 2009 and Cuzzucoli and Lombardo). Because damping force is proportional to velocity and higher modes move faster at a given amplitude, the finger removes high-frequency energy faster than a rigid boundary, but the sin² factor makes the filter position-dependent and comb-like, and real fingertips have finite patches whose stiffness and dissipation change with pressure. The one damping figure in the legs (about 2 Ns/m at 2 N, flesh stiffness 41.6 to 237 N/m, Hajian and Howe 1997 via a McGill thesis) is a haptics measurement of a fingertip against a probe, second-hand and unopened, entering only as an order of magnitude. A finger behind an established fret couples through that boundary; lifting the string off the fret and leaving flesh contact removes the hard node and the string dies fast. On a solidbody the soundboard loss channel is largely absent, so intrinsic and contact damping rise in relative weight, the structural reason fretting-hand damping is more audible on an electric (inference; the partition is UNVERIFIED). Intrinsic damping already varies by an order of magnitude across the band, so no single constant describes the added loss, and a changed composite envelope does not prove changed damping without partial-by-partial decay measurement. Class: MODEL; no string measurement of fingertip damping was opened.

A viscous point load extracts power proportional to the squared velocity of the string at the contact, which for mode n scales with sin²(nπx_f/L) and with frequency; where the structure drains little energy, the contact becomes a leading loss term.

• S1Woodhouse, Euphonics: Guitar I (plucked-string model, two polarizations, endpoint mobility; site 2022)MODEL: the stiff damped string equation, the triangular-pluck modal law, the two-polarization pluck-direction synthesis and the near-end transverse stiffness F ≈ Th(1/a + 1/(L−a)).
• S2Paté, Le Carrou, Fabre, Predicting the decay time of solid body electric guitar tones, JASA 135(5):3045 to 3055, 2014INTERVENTION: polarization-resolved decay of electric-guitar strings on a rigid fixture with two optical sensors, separating intrinsic string loss from structural coupling.
• S6Evangelista and Eckerholm, Player-Instrument Interaction Models for Digital Waveguide Synthesis of Guitar Touch and Collisions, IEEE TASLP, 2009MODEL: the finger or plectrum as a damped spring-mass system with time-varying parameters, fret collisions and imperfect stopping.
• S20Kry, PhD thesis, McGill (reports Hajian and Howe 1997, J. Biomech. Eng. 119(1), fingertip impedance)ARGUMENT (second-hand): fingertip damping about 2 Ns/m at a 2 N interaction force and stiffness 41.6 to 237 N/m, scaling nonlinearly with force; the primary record was not opened.
• S23Cuzzucoli and Lombardo, Physical model of the plucking process in the classical guitar (ICMC 1997; CMJ 1999)MODEL with empirically measured parameters: plucking force 1 to 5 N, exciter mass 0.5 to 3 g, finger stiffness up to about 3,000 N/m, displacement about 1 mm (piano) to 5 mm (forte) on one classical guitar.
• S26Woodhouse, Euphonics: Guitar IIMODEL + measurement: intrinsic string damping varies by an order of magnitude across the audible range, so no single decay constant describes a gesture's effect.

Bending lengthens the sounding span slightly (lowering pitch alone) but raises tension by EA(1/cos θ − 1), and tension wins, so the fundamental f ≈ (1/2L)√{[T + EA(1/cos θ − 1)]/(μ0 cos θ)} rises for a bend up or down alike; only magnitude matters. Grimes (2014) measured this on an electric guitar with nails driven into the neck holding strings at fixed bend angles across several strings. Near straight the path-length gain is quadratic, Δℓ ≈ (b²/2)(1/ℓ1 + 1/ℓ2), so with f − f0 ≈ Cb² a hand motion b0 + b1 sin Ωt yields a component at Ω proportional to 2b0b1 and one at 2Ω: around the unbent note (b0 = 0) pitch rises twice per hand cycle and never falls below f0; around a pre-bend the response is near-linear and bipolar at the hand rate. Hand rate, pitch-modulation rate and pitch slew are different quantities and the modulation is not sinusoidal in general. Axial vibrato (rolling along the string at zero bend angle) modulates tension directly and can raise and lower pitch. Frequency modulation puts sidebands around partials that are not new modes, and co-varying fingertip damping adds amplitude modulation. Friction at nut and saddle strands tension imbalances after a bend (Kemp 2017), so the same finger position need not reproduce the same pitch. Benito and McPherson's TMR sensor (2021) reads horizontal displacement on an electric guitar and calibrates it to bend pitch empirically, disambiguating up from down bends that sound alike. Grimes's vibrato treatment is a simulation and players differ widely in width and rate; circulating ranges (4 to 7 Hz, 40 to 140 cents) come from an unrefereed calculator and carry no number. Class: INTERVENTION (fixed-displacement bend pitches) + MODEL (vibrato), CORRELATIONAL sensing.

Lateral displacement stretches the string against its axial stiffness EA, and because frequency goes as √T the gain in tension outruns the loss from added length; the stretch is even in displacement, so an oscillation centered on zero is rectified to twice the drive frequency.

• S4Kemp, The physics of unwound and wound strings on the electric guitar applied to the pitch intervals produced by tremolo/vibrato arm systems, PLOS ONE, 2017-09-21INTERVENTION + MODEL: the string slips over nut and saddles to equalise tension within the limits of friction, leaving imbalanced tensions and tuning problems after a tremolo excursion.
• S7Grimes, String Theory: The Physics of String-Bending and Other Electric Guitar Techniques, PLOS ONE 9(7):e102088, 2014-07-23INTERVENTION (bend pitches measured at fixed displacements on an electric guitar) + MODEL (fretting-force equation, action dominance, radial and axial vibrato simulation, whammy-bar tension).
• S11Perez-Carrillo, Finger-String Interaction Analysis in Guitar Playing With Optical Motion Capture, Frontiers in Computer Science, 2019-11-15CORRELATIONAL (gesture capture): 240 Hz optical motion capture plus audio estimates contact, pressure and release phases, position, velocity, direction and displacement; no force measured.
• S22Benito Temprano and McPherson, A TMR Angle Sensor for Gesture Acquisition and Disambiguation on the Electric Guitar, Audio Mostly 2021 (DOI 10.1145/3478384.3478427)CORRELATIONAL (sensing): horizontal string displacement sensed on an electric guitar and calibrated empirically to bend pitch; upward and downward bends that sound alike are disambiguated.

Trace the forward signal in a passive guitar into a classic tube amplifier (the Fender 5F1 schematic, opened). The joints are: hand to string (mechanical); string to magnetic field (a change of flux linkage across an air gap, no conductor); field to coil (Faraday induction); coil through pots, jack and cable to the first grid (one conductive loop, A); grid to plate at each tube (an electrostatic field controlling a cathode-to-plate stream powered by the amplifier's own supply; the pickup's current never becomes the plate current); plate to the next grid through a coupling capacitor (displacement current across a dielectric, DC blocked); power tube to transformer primary (conductive loop B); primary to secondary (mutual induction, no copper path); secondary to voice coil (conductive loop C); voice coil to cone (Lorentz force F = Bl·i); cone to air to ear (acoustic). At least five joints have no ohmic path. What runs nearly end to end is the ground reference: strings and bridge bonded to the guitar common, common down the cable shield to the chassis, chassis to the speaker return; the 5F1 also carries a 22 kΩ feedback resistor from the secondary to a preamp cathode, so the amplifier is not 'isolated by the transformer' either. A continuity meter from a string to a speaker terminal may read a closed path; that is the reference route, not the audio's route. Bounds: holds for every passive-pickup, output-transformer tube chain with a grounded secondary; an OTL amplifier removes one magnetic boundary and keeps the vacuum and dielectric ones; active pickups and pedals add powered stages and more non-conductive joints, never fewer. Evidence class: MODEL (physics derivation) plus an opened schematic; population: a physical artifact, not a human sample; outcome: a topology count.

Faraday's law generates EMF from changing flux linkage with no conductor between string and coil; Maxwell's displacement-current term carries AC across a capacitor's dielectric while charges never cross it; mutual induction moves energy between transformer windings through shared flux, not shared copper; a triode's plate current is a locally powered stream controlled by the grid-cathode field. Energy is not conserved along the chain (each stage injects supply energy); the continuous-valued waveform is. 'Analog' names continuity of the signal's value in time and amplitude, never continuity of a wire.

• S1OpenStax University Physics Vol. 2, 16.1 Maxwell's Equations and Electromagnetic WavesDisplacement current carries AC across a capacitor's dielectric with no conduction; Faraday induction needs changing flux linkage, not a conductor
• S2OpenStax University Physics Vol. 2, 15.6 TransformersTransformer secondary voltage arises from shared changing flux, not conduction; reflected impedance scales with the square of the turns ratio
• S3Fender Champ 5F1 schematicA minimal tube chain: two triode stages, interstage coupling capacitors, power tube, output transformer with grounded secondary and a 22 kΩ feedback path to a preamp cathode
• S4JJ Electronic ECC83S datasheetTransconductance, plate resistance and interelectrode capacitances (1.6 pF grid-cathode, 1.7 pF grid-anode) used for the Miller input-capacitance estimate
• S5All About Circuits, Input and Output CouplingCapacitor and transformer coupling pass AC and block DC between amplifier stages and between an amplifier and its load
• S6StewMac, Understanding Guitar Wiring Part 8: Grounding and ShieldingIn a passive guitar the bridge, strings, pot cases, jack sleeve and shielding are bonded to one common ground
• S7Texas Instruments, Isolation technologiesGalvanic isolation defined: no direct conduction path, signal exchanged by capacitive, inductive or optical coupling
• S8Wikipedia, Output transformerlessThe OTL case: removing the output transformer removes one magnetic boundary but not the vacuum and dielectric ones

With a stationary coil the induced EMF is e(t) = −dλ/dt, λ the linked flux; the vibrating ferromagnetic string modulates λ, and the alternating flux is about 1 percent of the static field even for strong vibration (Lemme). Expanding λ about rest gives a leading term proportional to string velocity projected onto the pickup's sensitivity direction plus higher-order terms scaling with position times velocity. So: (a) the pickup reports velocity, displacement differentiated, weighting every partial by its own frequency, a +6 dB per octave tilt relative to a displacement sensor that no winding, magnet or pot undoes (confirmed from the other side by a position-sensing pickup patent claiming 6 dB per octave more low end); (b) because flux versus pole-to-string distance follows a hyperbolic curve, sensitivity changes with instantaneous position, modelled as a static nonlinearity followed by a derivative (DAFx 2018; Paiva, Pakarinen, Välimäki), so the chain's first distortion is the magnetic field's shape, not a tube; (c) the Guadagnin et al. bench, a controlled harmonic-motion rig driving a string segment past a pickup, reports velocity-sensitive behaviour and greatest sensitivity to motion normal to the pickup face in the tested arrangement, INTERVENTION-class evidence for that rig, not a universal constant. Height is a gain and a filter: near 3 mm, output falls about 3 dB per millimetre of added magnet-to-string distance (Zollner, measured on the single coils he swept). Population: physical pickups on a bench; outcome: measured voltage and spectrum; design: controlled physical measurement and physical model.

Faraday's law makes EMF proportional to the time derivative of linked flux; a pole piece's field falls off roughly hyperbolically with distance, so the displacement-to-flux map is curved, and differentiating a curved map of a sinusoid yields harmonics (a symmetric lateral map yields a term at twice the motion frequency). Lenz's law makes the transducer bilateral: the static magnet pulls on the string and loads its vibration.

• S9Guadagnin et al., AES e-library paper 19201 (controlled string-motion bench and pickup transduction model)Velocity-sensitive behaviour and greatest sensitivity to z-polarised motion measured on a controlled harmonic-motion rig
• S10Helmuth Lemme, The Secrets of Electric Guitar PickupsAlternating flux about 1 percent of static field; hyperbolic flux-distance curve; L 1 to 10 H, coil C 80 to 200 pF, cable C 300 to 1,000 pF; tone-down resonances near 700 and 450 Hz; magnet pull on the string; inductance rises quadratically with turns
• S11DAFx 2018 paper 39, nonlinear guitar pickup modellingMagneto-electric conversion modelled as a static nonlinearity followed by a derivative; output proportional to velocity times the gradient of the nonlinear flux map
• S12Paiva, Pakarinen, Välimäki, Acoustics and Modeling of PickupsDistance-dependent magnetic nonlinearity; the tone branch interacting with pickup inductance to form a lower resonance; exciter-core loading caution
• S13Zollner, Physik der Elektrogitarre ch. 5.4.1 (static magnetic field), English 2019About 3 dB output drop per mm of added magnet-string distance near 3 mm; resonance placement bounds (above 5 kHz barely noticeable, near 3 kHz ice pick, below 1 kHz muddy); damping required
• S14US Patent 9,679,549 (position-sensing string pickup)A position-sensing design claims 6 dB per octave more low-frequency response than a velocity pickup, confirming the velocity tilt from the other direction
• S15Horton and Moore, Rollins College, modelling the electric guitar pickupExperimental modelling of how the ferromagnetic string modifies the pickup's field and flux linkage

For an ideal fixed-end string of vibrating length ℓ, a linear pickup with sensitivity kernel g(x) outputs e(t) = −∫ g(x) u̇(x,t) dx, so partial n carries three separable factors: A_n, what the hand put into the mode; ω_n, the velocity tilt; and ∫ g(x) sin(nπx/ℓ) dx, where the pickup looks. For a point sensor at x_p the factor is sin(nπx_p/ℓ): every partial with a node over the pickup is absent from the linear output, and the nulls sit at F_node = (L_scale / X_pickup) · F_open, the pitch of the string fretted at the pickup location (Tillman). Worked on a 25.5 inch Stratocaster, low E 82.4 Hz: neck pickup at 6.375 inches from the bridge, first null 330 Hz (the 4th partial) and every multiple; middle at 3.875 inches, first null 542 Hz; bridge at 1.625 inches, first null 1274 Hz. On a 24.75 inch Les Paul, neck at 6.25 inches nulls first near 326 Hz, bridge at 1.5 inches near 1360 Hz (same formula). A finite aperture of width w adds a sinc(nπw/2ℓ) factor, a second higher comb; Zollner measures the magnetic window at about 1 inch for a thin single coil and 2.5 inches for a humbucker, and a series humbucker samples at two points, adding a third comb by the shift theorem. Bounds: Zollner's caution that because a single coil's electrical resonance limits transmission to about 5 kHz the aperture term has little influence on single coils, so position and electrical resonance dominate there; fretting changes ℓ so a fixed pickup does not keep its node relationships; an electrical resonance cannot recreate a partial the linear spatial sampling cancelled. Evidence class: MODEL (analytic, with checkable null predictions) plus Zollner's measured apertures; population: physical instruments.

A standing wave has fixed nodes; a sensor integrating motion over a region weights each mode by the mode shape's integral over that region. Sampling theory makes a point sample a sine comb and a rectangular window a sinc low-pass, and the two factors separate (Tillman, Zollner). The position comb and the plucking-point comb are the same mathematics at two points on the string.

• S16J. Donald Tillman, Response Effects of Guitar Pickup Position and Width (2000, updated 2002)Position comb derivation; nulls at the fretted-at-the-pickup pitch; Strat worked values (neck first null 330 Hz, middle 542 Hz, bridge 1274 Hz); width as a second separable comb
• S17Tillman, Pickup Response DemoInteractive demonstration of the position and width comb responses
• S18Zollner, Physik der Elektrogitarre ch. 5.4.4 (the magnetic aperture), English 2019Aperture about 1 inch for a thin single coil and 2.5 inches for a wide humbucker; aperture has little influence on single-coil transmission because the electrical resonance limits it to about 5 kHz; a pickup is not a flat measuring device
• S19Tillman, Pickup MixingThe two-pickup and two-coil sampling case and its comb
• S20AES e-library paper 16551 (pickup sensing aperture and polarisation weighting)The pickup's magnetic sensitivity occupies a region along the string and weights the two transverse polarisations differently

Electrically a passive pickup is a source EMF behind R_s + sL feeding a parallel load Y = 1/R_V + 1/Z_in + s(C_p + C_c) + the tone branch, so H(s) = 1 / (1 + (R_s + sL) Y(s)). With one parallel R_L and one total C it is a second-order low-pass: flat below, 12 dB per octave above, a peak near 1/(2π√(LC)) of height Q ≈ R_L √(C/L) (voltage peak slightly below the LC frequency, present only when Q > 1/√2). Measured magnitudes: L 1 to 10 H (2 to 3 H Strat single coil, 4 to 8 H humbucker); coil C about 80 to 200 pF (90 to 120 pF Strat, about 50 pF PAF, 250 to 350 pF single-coil-sized humbucker); cable C 300 to 1,000 pF for 10 to 20 feet at 70 to 150 pF per metre (Belden 9778: 45 pF per foot, 148 pF per metre, so 20 feet is 900 pF; page revised 2026-02-20). The cable's C is typically three to ten times the coil's, so the cable, not the pickup, usually sets the resonant frequency. Lemme 2013: a 1972 bridge pickup, 2.2 H, under 470 pF and 200 kΩ peaks near 4.5 kHz. Macalister: a Strat pickup through its cable shows an impedance maximum at 4.33 kHz, inferred R 8.61 kΩ, C 579 pF, L 2.33 H (2.41 H corrected). Bench: a Strat pickup peaks near 8 kHz with 47 pF and near 2.4 kHz with 2,200 pF added; a 1960s 8 to 10 foot cable at about 300 pF put a 2.4 H pickup near 6 kHz while a 2,400 pF coiled cord dropped it just above 2 kHz, two octaves from a cable. Resistance is the other knob: the peak must be damped or a pickup 'sounds terrible'; above 5 kHz barely noticeable, near 3 kHz the 'ice pick', below about 1 kHz muddy (Zollner); pots load in parallel, 500 kΩ volume plus 500 kΩ tone presents 250 kΩ, about 200 kΩ with a 1 MΩ amplifier. The openai-dr model table (L 2.4 H, R_s 8.6 kΩ, C_p 100 pF assumed): cable C 200 to 900 pF moves f from 5.93 to 3.25 kHz while Q rises 1.88 to 2.43; at fixed 600 pF, R_L 1 MΩ to 111 kΩ drops Q 5.04 to 1.47 at 4.19 kHz. More capacitance lowers the peak, it does not merely flatten treble. The amplifier input is the last parallel element and not purely resistive: a 12AX7's Miller input capacitance C_gk + C_ga(1 − A_v) is about 88 pF at an assumed gain of −50 from datasheet values 1.6 and 1.7 pF. A buffer right after the pickup is the one way to make the response cable-independent; capacitance before the buffer still loads the coil. Bounds: lumped L, C and resistive load are models; eddy-current losses in covers and poles and the distributed winding make real coils lossier; DC resistance alone predicts neither output nor response. Evidence class: INTERVENTION (bench sweeps under stated loads) and MODEL; population: physical pickups and cables; outcome: measured impedance or voltage-response curves.

A coil's inductance stores energy magnetically and the cable's and coil's capacitance store it electrically; they exchange energy at one frequency where the source's reactance peaks and output rises. A resistive shunt bleeds that energy each cycle, so resistance sets the peak's height while the LC product sets its place. Coaxial capacitance per length is 2πε/ln(b/a), dielectric and geometry only; at audio frequencies the cable's wavelength is tens of kilometres, so it is a lumped capacitor, never a transmission line.

• S10Helmuth Lemme, The Secrets of Electric Guitar PickupsAlternating flux about 1 percent of static field; hyperbolic flux-distance curve; L 1 to 10 H, coil C 80 to 200 pF, cable C 300 to 1,000 pF; tone-down resonances near 700 and 450 Hz; magnet pull on the string; inductance rises quadratically with turns
• S21Helmuth Lemme, Pickup Measuring Techniques (2013 article, hosted PDF)A 1972 bridge pickup of 2.2 H under 470 pF and 200 kΩ peaks near 4.5 kHz; intermediate volume settings suppress the peak; bypass capacitors change the curve shape; the 470 pF measurement convention
• S22Macalister Electronics, Guitar Pickup Equivalent CircuitsJack-impedance sweep of a Strat pickup: impedance maximum 4.33 kHz, inferred R 8.61 kΩ, C 579 pF, L 2.33 H (2.41 H with damping correction); measuring apparatus is part of the network
• S23Belden 9778 analog audio cable datasheet (revised 2026-02-20)45 pF per foot (about 148 pF per metre), so a 20 foot cable adds about 900 pF
• S13Zollner, Physik der Elektrogitarre ch. 5.4.1 (static magnetic field), English 2019About 3 dB output drop per mm of added magnet-string distance near 3 mm; resonance placement bounds (above 5 kHz barely noticeable, near 3 kHz ice pick, below 1 kHz muddy); damping required
• S24Rafael Paiva, Circuit modeling studies related to guitars and audio processing (thesis summary)Cables measured alone and with a pickup; the distributed cable model reduces to a lumped capacitance that dominates the resonance shift; a buffer replaces the driving impedance but capacitance before it still loads the coil
• S25arXiv 1508.01246 (guitar pickup equivalent circuit modelling)Effective terminal capacitance is a lumped representation of distributed winding capacitance
• S26Batchelor et al., Guitar Pickups I: Analysis of the Effect of Winding and Wire Gauge on Single Coil Electric Guitar Pickups (2024)Dedicated pickups varying only turns (5,000 to 12,000) at 42 and 44 AWG, impedance swept: resonant frequency falls exponentially with turns, peak magnitude rises linearly; R_dc proportional to N, L proportional to N squared
• S27Sound on Sound, Choosing Guitar CablesTypical guitar-cable capacitance per foot and its scaling with length
• S4JJ Electronic ECC83S datasheetTransconductance, plate resistance and interelectrode capacitances (1.6 pF grid-cathode, 1.7 pF grid-anode) used for the Miller input-capacitance estimate

Write the volume pot as R_a = (1 − α)R_V in series toward the pickup and R_b = αR_V to ground, α the electrical wiper position. The cable and amplifier input load the wiper node, so V/E = [Z_A / (Z_s + Z_A)] · [Z_B / (R_a + Z_B)], a two-stage divider whose impedances all depend on frequency. At full rotation the pot is a parallel resistance setting Q; at any lower setting the series arm raises the source impedance the cable sees and forms an RC low-pass with the cable's capacitance (a 250 kΩ pot's series arm reaches about R_V/4 at mid-rotation, which with 500 pF gives a corner near 5.1 kHz in the simplified low-source-impedance case). Lemme's measured curves under 470 pF show intermediate volume settings substantially suppressing the main peak rather than shifting the full-volume curve down; a treble-bypass capacitor across the series arm yields another curve shape, and halving the pot value requires halving the bleed resistor and doubling the bleed capacitor. The tone control is a series capacitor to ground through the tone pot: at '10' the pot isolates the capacitor so it loads only at high frequency; at minimum the capacitor sits across the pickup, dwarfs the cable's capacitance (22,000 pF against 600 pF) and drops the LC frequency to about 680 Hz with 0.022 µF or 470 Hz with 0.047 µF at L = 2.4 H (Lemme states near 700 and 450 Hz). Bounds: standard passive Fender and Gibson wiring with the tone branch at the pickup-side node; a relocated tone branch or a treble-bleed changes the network; the 5.1 kHz figure is a model illustration, not a measured knob response; pot taper and tolerance set the actual curve. Evidence class: INTERVENTION (Lemme's measured sweeps at stated load and settings) and MODEL; population: a passive guitar circuit on a bench; outcome: a measured frequency response.

A potentiometer is a divider whose upper arm is in series with a capacitive load, so its ratio depends on frequency wherever the source impedance is not negligible; a passive pickup's source impedance is tens of kilohms and rises with frequency through its inductance, so no passive volume setting below maximum is frequency-neutral. A capacitor across an inductor relocates their shared resonance to 1/(2π√(LC)); the tone capacitor is a very large C added to the same tank as the cable.

• S21Helmuth Lemme, Pickup Measuring Techniques (2013 article, hosted PDF)A 1972 bridge pickup of 2.2 H under 470 pF and 200 kΩ peaks near 4.5 kHz; intermediate volume settings suppress the peak; bypass capacitors change the curve shape; the 470 pF measurement convention
• S10Helmuth Lemme, The Secrets of Electric Guitar PickupsAlternating flux about 1 percent of static field; hyperbolic flux-distance curve; L 1 to 10 H, coil C 80 to 200 pF, cable C 300 to 1,000 pF; tone-down resonances near 700 and 450 Hz; magnet pull on the string; inductance rises quadratically with turns
• S22Macalister Electronics, Guitar Pickup Equivalent CircuitsJack-impedance sweep of a Strat pickup: impedance maximum 4.33 kHz, inferred R 8.61 kΩ, C 579 pF, L 2.33 H (2.41 H with damping correction); measuring apparatus is part of the network
• S12Paiva, Pakarinen, Välimäki, Acoustics and Modeling of PickupsDistance-dependent magnetic nonlinearity; the tone branch interacting with pickup inductance to form a lower resonance; exciter-core loading caution

Batchelor, Gooding, Marriott, Chalashkanov, Tucker and Margetts (Nottingham Trent and Lincoln, arXiv 2409.19782, 2024) built dedicated Stratocaster-style single coils differing in one parameter at a time, turns across the commercial range 5,000 to 12,000 at 42 and 44 AWG, and swept each on a frequency response analyser for impedance. The abstract reports resonant frequency decreasing exponentially with turns and resonant peak magnitude increasing linearly with turns; the numerical table sits behind the abstract the legs opened. The winding model explains the direction: for fixed geometry L is proportional to N squared and R_dc to N, and the EMF depends on which turns link the string's changing flux, so adding turns raises generation and inductive loading together. Two coils of equal DC resistance on 42 versus 43 or 44 AWG carry different turn counts, inductances and resonances, which is why the industry's DCR figure measures neither output nor tone. Bounds: measured on Stratocaster-style single coils with the study's magnets and bobbin, as an impedance sweep rather than a string-driven output; range 5,000 to 12,000 turns; metal covers, steel poles and eddy-current losses in humbuckers add frequency-dependent loss the N-scaling omits; 'exponentially' is the authors' abstract wording and the exact exponent was not opened. Evidence class: INTERVENTION (one parameter varied, the rest held); population: bench-built pickups; outcome: impedance versus frequency.

A coil's inductance scales with the square of turns because each turn both produces flux and links every other turn's flux; resistance scales with wire length and inversely with wire area. With winding capacitance roughly fixed by the bobbin, the LC frequency falls about as 1/N, and against a fixed load the peak grows because the coil's reactance at resonance grows with L.

• S26Batchelor et al., Guitar Pickups I: Analysis of the Effect of Winding and Wire Gauge on Single Coil Electric Guitar Pickups (2024)Dedicated pickups varying only turns (5,000 to 12,000) at 42 and 44 AWG, impedance swept: resonant frequency falls exponentially with turns, peak magnitude rises linearly; R_dc proportional to N, L proportional to N squared
• S28Batchelor et al., Guitar Pickups I, ar5iv HTML rendering (2024)The same controlled turn-count study's abstract text as opened by the exa leg
• S10Helmuth Lemme, The Secrets of Electric Guitar PickupsAlternating flux about 1 percent of static field; hyperbolic flux-distance curve; L 1 to 10 H, coil C 80 to 200 pF, cable C 300 to 1,000 pF; tone-down resonances near 700 and 450 Hz; magnet pull on the string; inductance rises quadratically with turns
• S22Macalister Electronics, Guitar Pickup Equivalent CircuitsJack-impedance sweep of a Strat pickup: impedance maximum 4.33 kHz, inferred R 8.61 kΩ, C 579 pF, L 2.33 H (2.41 H with damping correction); measuring apparatus is part of the network

At the input end Lenz's law and the static field make the transducer two-way: the pickup's permanent magnet exerts a steady attraction on the ferromagnetic string that alters its vibration, which is why very strong magnets or a pickup set too close are avoided (Lemme; Zollner's static-field chapter). At the output end a moving-coil loudspeaker obeys F = Bl·i for force and e_back = Bl·v for the voltage the moving coil generates (Bl the force factor, Klippel); its terminal impedance is Z = R_e + sL_e + (Bl) squared / Z_m(s), so the cone's mechanical resonance and the air load appear electrically at the terminals (Kraftmakher's loudspeaker experiments connect cone motion, back-EMF and the measured impedance peak). Through the output transformer that impedance is reflected to the tubes as Z_reflected = (N_p/N_s) squared times Z_speaker, so the plate sees a load rising at the speaker's resonance and above a few kilohertz as voice-coil inductance grows. The back-EMF opposes the motion that creates it and is dissipated through the amplifier's output impedance: the lower that impedance, the more the back-EMF current brakes the cone. A pentode or beam-tetrode push-pull stage with little or no negative feedback has an output impedance comparable to or above the speaker's own, so the cone is less braked than on a low-impedance solid-state output and the speaker's resonances ring longer; this is the inference, and the Fender 5F1's 22 kΩ feedback resistor from the secondary to a cathode is the schematic evidence that designers add feedback to lower that output impedance. Bounds: the bilateral laws hold for every magnetic pickup and moving-coil speaker; the under-damping magnitude depends on the specific amplifier's output impedance, which no leg could open for a named guitar amplifier (anthropic's search 'loudspeaker voice coil force factor Bl back EMF impedance curve damping' returned general loudspeaker theory only); Bl(x) is position-dependent, a loudspeaker nonlinearity in Klippel's list. Evidence class: MODEL (transducer theory) with opened laboratory demonstrations (Kraftmakher) and one opened schematic; population: physical transducers; outcome: measured impedance and back-EMF.

Reciprocity: any transducer converting current to force converts velocity to voltage with the same coupling coefficient Bl, so the mechanical side's impedance appears on the electrical side scaled by Bl squared. A source with high internal impedance cannot sink the back-EMF current, so it cannot brake the cone; negative feedback lowers a tube amplifier's output impedance by sensing the secondary voltage and correcting the drive.

• S29Klippel, Loudspeaker Nonlinearities: Causes, Parameters, SymptomsForce factor Bl as the coupling between mechanical and electrical sides; back-EMF u = Bl(x)·v; Bl(x) position dependence as a nonlinearity
• S30Kraftmakher, Loudspeaker experiments, European Journal of Physics 31 (2010)Laboratory measurements connecting cone motion, back-EMF and the loudspeaker's electrical impedance peak at mechanical resonance
• S31AudioHeritage attachment on loudspeaker damping and back-EMFA moving cone acts as a generator; its back-EMF damps the motion through the amplifier's output circuit, and the lower that impedance the more the back-EMF controls the cone
• S3Fender Champ 5F1 schematicA minimal tube chain: two triode stages, interstage coupling capacitors, power tube, output transformer with grounded secondary and a 22 kΩ feedback path to a preamp cathode
• S2OpenStax University Physics Vol. 2, 15.6 TransformersTransformer secondary voltage arises from shared changing flux, not conduction; reflected impedance scales with the square of the turns ratio
• S32Amperatubes, Output Transformers (2024-09-01)Z_primary = N squared times Z_load; an EL34 wants about 3.4 kΩ against an 8 Ω speaker; energy crosses by coupling through a shared iron core at 85 to 95 percent efficiency
• S10Helmuth Lemme, The Secrets of Electric Guitar PickupsAlternating flux about 1 percent of static field; hyperbolic flux-distance curve; L 1 to 10 H, coil C 80 to 200 pF, cable C 300 to 1,000 pF; tone-down resonances near 700 and 450 Hz; magnet pull on the string; inductance rises quadratically with turns
• S13Zollner, Physik der Elektrogitarre ch. 5.4.1 (static magnetic field), English 2019About 3 dB output drop per mm of added magnet-string distance near 3 mm; resonance placement bounds (above 5 kHz barely noticeable, near 3 kHz ice pick, below 1 kHz muddy); damping required

For a smooth, memoryless, asymmetric gain stage (a common-cathode triode is the measured case), the nth harmonic relative to the fundamental scales as about the (n-1)th power of input amplitude while drive stays well under the stage's headroom, so second order dominates at small drive and third order takes over as drive approaches the knee. Measured: Zollner's input stage of a guitar amp reaches about 3 percent harmonic distortion (harmonic-to-output voltage ratio, one ECC83 stage, 250 V supply) at 300 mV rms input, with 2nd order predominant at small input and 3rd order dominant from 0.25 to 1 V, the border depending on grid resistor, supply voltage and tube specimen (S2, 2007). Dempwolf and Zölzer measured a common-cathode stage at 500 Hz with 2, 4 and 8 V input and found progressive limiting with both even and odd harmonics (S1, 2011). Keeports drove a valve preamp with a 200 Hz tone: moderate overdrive showed prominent 2nd and 4th harmonics and a weak 3rd, and raising gain strengthened the 3rd and weakened the 4th (S16, 2017, one amplifier). The load line, not the tube alone, sets the transfer (S17). Boundary: a single-ended stage below grid conduction and above cutoff; the power series is an asymptote, and real tubes' perveance and amplification factor vary with operating region (S1). Population: bench stages and one amplifier; outcome: harmonic spectrum against input level; evidence class: INTERVENTION with MODEL fit. Says nothing about the power stage (A6) or memory (A4, A5).

Expand y = a1 x + a2 x^2 + a3 x^3 about the bias point with x = A cos(wt): B2/B1 is proportional to A and B3/B1 to A^2. The space-charge three-halves law gives a2 > 0 and a3 < 0 for a triode, so the even term leads and the odd term both compresses the fundamental and grows faster; an asymmetric curve (soft cutoff one way, grid conduction the other, S15) is what admits even terms at all.

• S1Dempwolf and Zölzer, A physically-motivated triode model for circuit simulations, DAFx-11 (2011)A common-cathode triode stage measured at 500 Hz with 2, 4 and 8 V input shows progressive limiting with both even and odd harmonics; the circuit model reproduces most but not all of it (INTERVENTION + MODEL; fetched 2026-10-05).
• S2Zollner, Physics of the Electric Guitar, ch. 10.1 Guitar amplifiers (2007; English edition)The input stage of a guitar amp reaches about 3 percent harmonic distortion at 300 mV rms, 2nd order predominant at small input and 3rd order dominant from 0.25 to 1 V, the border set by grid resistor, supply voltage and tube specimen; small-signal gain -42; a Fender Deluxe supply fell from 247 V to 210 V under full drive from 0.3 to 1.7 s (INTERVENTION).
• S16Keeports, The warm, rich sound of valve guitar amplifiers, Physics Education 52 025010 (2017)A 200 Hz tone into a moderately overdriven valve preamp shows prominent 2nd and 4th harmonics and a weak 3rd; raising preamp gain strengthens the 3rd and weakens the 4th, a shift from even toward odd with drive (INTERVENTION, one amplifier, one experimenter).
• S17Blencowe (Valve Wizard), The Common Gain Stage (undated)The stage's input-to-output transfer is the intersection of the valve characteristic with the circuit's load line; bias, plate load, source impedance and cathode feedback set the usable excursion (ARGUMENT).
• S15Rutt, Vacuum Tube Triode Nonlinearity as Part of the Electric Guitar Sound, AES 76th Convention preprint 2141 (1984)Grid limiting is a soft limiting effect at the upper end of the load line; a class A triode stage direct-coupled to a cathode follower limits both the positive and negative sides of the guitar waveform (MODEL, load-line analysis with oscilloscope traces).

Where the dominant nonlinearity near threshold is of order n, total harmonic distortion changes tenfold over 20/(n-1) dB of input: 20 dB for a second-order knee, 10 dB for a symmetric third-order knee, and only a few dB for a hard clipper whose local order is very high. Touch-controlled clean-up therefore requires a soft knee whose threshold sits inside the span of input levels the hands produce; a hard-knee stage anywhere in the chain (a diode-clipping pedal, a hard-limited stage) imposes its own narrow window on the whole chain whether the element is a valve, a diode or a transistor. The window is a MODEL result derived from the same power series A1 measures (S1, S8); the hard-clipper end is consistent with Yeh, Abel and Smith's measured pedal spectra, where measured circuits showed strong even-order content that symmetric clipping models did not reproduce (S14, 2007). Boundary: a derived bound, not a measured window on any complete amplifier; no leg opened a joint dataset of input amplitude by gain by guitar volume by pedal drive. Population: circuit models and bench pedal measurements; outcome: THD against input level; evidence class: MODEL with INTERVENTION support at the pedal. The hands' dynamic span in dB is not measured here (the hands on the string vector owns it).

THD is proportional to A^(n-1) in the weakly nonlinear region, so a decade of THD spans 20/(n-1) dB of input amplitude; as the knee hardens, the local order n rises and the span collapses toward the clip threshold itself.

• S1Dempwolf and Zölzer, A physically-motivated triode model for circuit simulations, DAFx-11 (2011)A common-cathode triode stage measured at 500 Hz with 2, 4 and 8 V input shows progressive limiting with both even and odd harmonics; the circuit model reproduces most but not all of it (INTERVENTION + MODEL; fetched 2026-10-05).
• S8Boyk and Sussman, Small-Signal Distortion in Feedback Amplifiers for Audio, Caltech (2003)Push-pull stages cancel even-order terms; in the crossover region relative distortion worsens for smaller excitations with and without feedback, 18 dB of feedback leaving some intermodulation products almost 20 dB worse, the effect disappearing at larger excitation (MODEL, circuit simulation).
• S14Yeh, Abel, Smith, Simplified, physically-informed models of distortion and overdrive guitar effects pedals, DAFx-07 (2007)Measured distortion pedals at 220 Hz and 200 mV show a strong even-order nonlinearity not reproduced by symmetric clipping models; a pedal distorts the waveform before any valve receives it (INTERVENTION + MODEL).

Softer input cleans the output only when it lowers the drive at every stage that was distorting (a pedal ahead of the amplifier, each cascaded preamp stage, the power stage, the speaker) and only after the chain's stored states (A4, A5, A8) have recovered. A gain control lowers drive only for stages downstream of it; a master volume after a clipped preamp quiets the distortion but does not remove it; a pedal whose own clipping stage stays driven delivers a distorted waveform a clean amplifier reproduces, and measured pedals generate strong even-order content before any valve sees the signal (S14). The supply's reaction is present at low settings: on the Bassman 5F6-A the screen-supply drop occurred at volume 2 of 10 and grew at 10 of 10 (S4). The guitar's volume control is not a scalar attenuator: turning it down raises the source impedance seen by the cable and shifts the pickup-cable resonance, so part of what is heard as clean-up by the guitar volume is treble loss (S18; the passive-circuit physics belongs to the pickup and the circuit vector). Boundary: a deduction from stage topology with measured support at the pedal and the supply; no measured input-by-gain-by-guitar-volume-by-pedal matrix was opened by any leg. Population: circuit deduction and bench measurements; outcome: which stage's distortion a given control changes; evidence class: MODEL with INTERVENTION support (S4, S14).

Stage j distorts when its drive A times the upstream transfer exceeds its present headroom H_j; attenuation applied at point k changes only the drives downstream of k, and nonlinearity already committed upstream is a new waveform, not a gain that a later control can reverse.

• S14Yeh, Abel, Smith, Simplified, physically-informed models of distortion and overdrive guitar effects pedals, DAFx-07 (2007)Measured distortion pedals at 220 Hz and 200 mV show a strong even-order nonlinearity not reproduced by symmetric clipping models; a pedal distorts the waveform before any valve receives it (INTERVENTION + MODEL).
• S4Erbes, Leontopoulos, Zollner, Audibility of different power supplies in a guitar amplifier, DAGA 2018Bassman 5F6-A: screen-supply voltage drops about 60 V under a guitar piece, the drop present at volume 2 of 10 and larger at 10 of 10, with attack compression and enhanced fade-out; ABX with 17 experienced listeners, 15 trials each, 255 per condition, threshold 144, scores 158, 147, 172, 140, three of four significant, effect called very small (INTERVENTION, bench + listening).
• S18Ironstone Pickups, Guitar impedance matching (LCR) (undated)Pickup inductance with cable and self-capacitance forms a resonant circuit; turning the volume pot down raises source impedance and loses treble, so clean-up by the guitar volume is partly a low-pass (ARGUMENT).
• S2Zollner, Physics of the Electric Guitar, ch. 10.1 Guitar amplifiers (2007; English edition)The input stage of a guitar amp reaches about 3 percent harmonic distortion at 300 mV rms, 2nd order predominant at small input and 3rd order dominant from 0.25 to 1 V, the border set by grid resistor, supply voltage and tube specimen; small-signal gain -42; a Fender Deluxe supply fell from 247 V to 210 V under full drive from 0.3 to 1.7 s (INTERVENTION).

When a driven grid conducts, its current charges the coupling capacitor and shifts the stage's bias toward cutoff; the shift decays only through the grid resistor's RC path, so a soft note immediately after a hard one is attenuated, distorted or silenced, the opposite of clean-up. Measured: on a Fender Super Reverb with the global negative-feedback loop deactivated, a 1 kHz tone overdriving the power amplifier from t = 0 and reduced by 20 dB at t = 100 ms made the loudspeaker voltage collapse for a short time (S3, Zollner Fig. 10.4.18, 2007); the same measurement shows saddle-point distortions at the zero crossings in overdrive that Zollner traces to grid-current potential shifts, not to bias setting or transformer saturation. The recovery mechanism (coupling capacitor charged by grid current, bleeding through the grid-leak resistor; the shift pushing a class AB stage toward class B) is the designer's account in S10; grid limiting is a soft limit and the triode plus direct-coupled cathode follower limits both polarities (S15, 1984). Boundary: one bench amplifier for the measured collapse, feedback disabled; Zollner says not to dramatize it (hearing's post-masking covers part) and not to ignore it, since time constants vary by design. Population: one measured amplifier plus designer tutorials; outcome: output envelope after a level step; evidence class: INTERVENTION (S3) with ARGUMENT (S10) and MODEL (S15). Severity depends on grid-stopper value, coupling capacitance and grid-leak resistance.

Charge conservation on the coupling capacitor: rectified grid current must go somewhere, and where it goes is the bias point; the release is the RC of capacitor and grid resistor, independent of the present input amplitude.

• S3Zollner, Physics of the Electric Guitar, ch. 10.4 Phase splitter and driver (2007; English edition)Fig. 10.4.18: Super Reverb with global feedback deactivated; a 1 kHz tone overdriving the power amp reduced by 20 dB at t = 100 ms makes the loudspeaker voltage collapse briefly; saddle-point distortions at zero crossings in overdrive trace to grid-current potential shifts, not bias or transformer saturation (INTERVENTION).
• S10Aiken Amplification, What is Blocking Distortion? (undated)Grid current under overdrive charges the coupling capacitor, shifting bias toward cutoff (bias excursion) and pushing a class AB stage toward class B; recovery bleeds through the grid resistor; blocking distortion is not clipping (ARGUMENT).
• S15Rutt, Vacuum Tube Triode Nonlinearity as Part of the Electric Guitar Sound, AES 76th Convention preprint 2141 (1984)Grid limiting is a soft limiting effect at the upper end of the load line; a class A triode stage direct-coupled to a cathode follower limits both the positive and negative sides of the guitar waveform (MODEL, load-line analysis with oscilloscope traces).

A high-impedance power supply (rectifier drop plus winding resistance against a reservoir capacitor) lets the rail fall when the output stage's average current rises, lowering headroom and gain during the note and restoring them as it decays; it changes an envelope rather than an instantaneous waveform, so it is compression, and it exists only where current draw rises with level (class AB; an ideal class A stage draws near-constant current, S9). Measured: a Fender Deluxe supply node fell from 247 V to 210 V while fully driven from t = 0.3 to 1.7 s, and maximum level and harmonic distortion changed with it (S2, 2007). On a Bassman 5F6-A with three power supplies, the screen-supply voltage dropped about 60 V with a slight delay under a guitar piece, the drop was present at volume 2 of 10 and larger at 10 of 10, and at 10 of 10 the output showed the attack peak reduced and the fade-out enhanced (S4, 2018). In the same study's ABX test, 17 listeners with prior audio experience (guitarists, music production or audio study) did 15 trials each, 255 trials per condition; against a Bonferroni-corrected threshold of 144 correct, the four conditions scored 158, 147, 172 and 140, three of four significant, and the authors call the effect very small and probably of little relevance in practice, doubting the low-noise comparison because a faint tone gave a cue (S4). Boundary: two amplifiers measured; one listening test over headphones with recorded stimuli at volume 2 of 10, trials not fully independent (the authors say so). Population: bench measurements and 17 experienced listeners; outcomes: rail voltage against time, output envelope, ABX discrimination; evidence class: INTERVENTION. The listening result is a discrimination of supplies, not a preference and not a measure of touch response.

C dB/dt = i_rectifier - i_load: a sustained rise in load current drains the reservoir through the supply's equivalent resistance with time constant R_eq C, and the rail recovers on the same constant when demand falls; the audible effect is whatever the falling rail does to gain, bias and available swing.

• S2Zollner, Physics of the Electric Guitar, ch. 10.1 Guitar amplifiers (2007; English edition)The input stage of a guitar amp reaches about 3 percent harmonic distortion at 300 mV rms, 2nd order predominant at small input and 3rd order dominant from 0.25 to 1 V, the border set by grid resistor, supply voltage and tube specimen; small-signal gain -42; a Fender Deluxe supply fell from 247 V to 210 V under full drive from 0.3 to 1.7 s (INTERVENTION).
• S4Erbes, Leontopoulos, Zollner, Audibility of different power supplies in a guitar amplifier, DAGA 2018Bassman 5F6-A: screen-supply voltage drops about 60 V under a guitar piece, the drop present at volume 2 of 10 and larger at 10 of 10, with attack compression and enhanced fade-out; ABX with 17 experienced listeners, 15 trials each, 255 per condition, threshold 144, scores 158, 147, 172, 140, three of four significant, effect called very small (INTERVENTION, bench + listening).
• S9Aiken Amplification, What is Sag? (undated)Tube-rectifier drop varies with current; class AB draws far more current at full output than at idle; sag requires level-dependent current and is weak in true class A (ARGUMENT).

Two matched halves driven in antiphase cancel even-order terms differentially, so a power stage's own distortion is odd-order and any even harmonic measured at its output is a balance measurement (S8; Zollner's k3 = 12.3 percent under the anti-phase condition, S2). At the zero crossing the distortion is fixed in size by the bias, so its share of the signal grows as the signal shrinks: Boyk and Sussman's simulated push-pull stage shows relative distortions worsening for smaller excitations both with and without feedback, 18 dB of feedback leaving some intermodulation products almost 20 dB worse than without it, the effect disappearing as excitation grows (S8, 2003). Zollner's Super Reverb measurement shows saddle-point distortions at the zero crossings in overdrive, caused by grid-current potential shifts rather than bias setting or transformer saturation (S3, 2007). Once the stage clips, loop gain collapses and the feedback that lowered output impedance stops acting, so the response against the speaker's impedance curve changes with level as well (S11, designer's account; one leg). Boundary: even-order cancellation is exact only for matched halves; the crossover result is a circuit simulation plus one bench trace; the output-impedance clause rests on an ARGUMENT source and one leg. Population: circuit models and one measured amplifier; outcome: harmonic and intermodulation spectrum against excitation size; evidence class: MODEL (S8) with INTERVENTION (S3). Idle bias and feedback depth set the notch's size.

y(u) = f(V_b + u) - f(V_b - u) has no even powers; a fixed dead zone of width d contributes a distortion whose relative size scales as d/A, the inverse of the clean-up law in A1.

• S8Boyk and Sussman, Small-Signal Distortion in Feedback Amplifiers for Audio, Caltech (2003)Push-pull stages cancel even-order terms; in the crossover region relative distortion worsens for smaller excitations with and without feedback, 18 dB of feedback leaving some intermodulation products almost 20 dB worse, the effect disappearing at larger excitation (MODEL, circuit simulation).
• S3Zollner, Physics of the Electric Guitar, ch. 10.4 Phase splitter and driver (2007; English edition)Fig. 10.4.18: Super Reverb with global feedback deactivated; a 1 kHz tone overdriving the power amp reduced by 20 dB at t = 100 ms makes the loudspeaker voltage collapse briefly; saddle-point distortions at zero crossings in overdrive trace to grid-current potential shifts, not bias or transformer saturation (INTERVENTION).
• S2Zollner, Physics of the Electric Guitar, ch. 10.1 Guitar amplifiers (2007; English edition)The input stage of a guitar amp reaches about 3 percent harmonic distortion at 300 mV rms, 2nd order predominant at small input and 3rd order dominant from 0.25 to 1 V, the border set by grid resistor, supply voltage and tube specimen; small-signal gain -42; a Fender Deluxe supply fell from 247 V to 210 V under full drive from 0.3 to 1.7 s (INTERVENTION).
• S11Aiken Amplification, What is Negative Feedback? (undated)Guitar amps typically use 6 to 10 dB of global feedback; feedback lowers output impedance and raises damping factor only while loop gain exists (ARGUMENT).

Core flux scales as primary voltage over frequency, so for a given output voltage the lowest notes drive the iron hardest; two guitar-amplifier output transformers measured by Paiva, Pakarinen, Välimäki and Tikander introduced distortion at low frequencies only, below about 100 Hz for the Fender NSC041318 and 30 Hz for the Hammond T1750V (S13, 2011). Across level the term is not monotonic: magnetic hysteresis raises low-frequency distortion at relatively low signal levels and saturation raises it at high levels, and the distortion falls rapidly with increasing frequency (S12, Jensen's measured curves). Playing softer therefore does not clean the transformer's low-frequency contribution and can raise its share; above the low band the transformer's effect is linear filtering (leakage inductance, winding resistance and capacitance), not distortion. Boundary: two transformers measured; Jensen's curves are a manufacturer's handbook chapter. Population: bench transformers; outcome: harmonic distortion against frequency and level; evidence class: INTERVENTION (S13) and ARGUMENT with measured figures (S12). DC imbalance in the primary pre-biases the core and moves these thresholds (S12).

v = N A dB/dt gives B_pk = V_pk / (2 pi f N A); permeability falls near saturation and the B-H loop's hysteresis is a fixed-size nonlinearity whose relative size grows as the flux swing shrinks.

• S13Paiva, Pakarinen, Välimäki, Tikander, Real-Time Audio Transformer Emulation for Virtual Tube Amplifiers, EURASIP J. Adv. Signal Process. 2011:347645 (2011-02-22)Measured Fender NSC041318 and Hammond T1750V output transformers introduce distortion at low frequencies only, below about 100 Hz for the Fender and 30 Hz for the Hammond (INTERVENTION + MODEL; opened by the exa leg 2026-10-04; Springer answers a plain fetch with a JavaScript challenge).
• S12Whitlock, Audio Transformers (handbook chapter, Jensen Transformers; uploaded 2014)Magnetic hysteresis causes increased harmonic distortion for low-frequency signals at relatively low levels and saturation at high levels; distortion decreases rapidly with increasing frequency (ARGUMENT with measured curves).

A guitar loudspeaker measured at 30 W input, 1.5 m on axis, gave total harmonic distortion relative to the fundamental of -9 dB at 70 Hz (2nd -11, 3rd -13), -25 dB at 100 Hz, -36 dB at 140 Hz, -39 dB at 200 and 400 Hz and -37 dB at 4 kHz; the authors put THD near resonance at roughly 10 to 30 percent and in the high frequencies around 1 percent, and state that much of the salient timbre the loudspeaker imparts comes from its linear filtering (S6, 2008, one driver). Klippel's catalogue puts the low-frequency distortion on force factor Bl(x) and suspension stiffness Kms(x), with cone and surround nonlinearity appearing only after break-up, the break-up irregularities themselves being linear modal behaviour (S19). Fundamental compression is separable into thermal and motor causes: in Klippel's example driver swept 1 to 8 V rms, the fundamental was reduced by 7 dB below resonance, 5 dB at resonance and about 2 dB above it relative to a linear system, the current-based measure isolating the thermal part (S5). On a guitar loudspeaker driven at 20 W average and 60 V peak, voice-coil resistance varied by more than 15 percent, the amplitude response varied 1 to 2 dB within a few seconds from heating, and excursion was asymmetric (+4 mm against -3 mm, a dynamic DC displacement) (S7). Heat releases over seconds, so thermal compression persists into the soft note after a loud passage. Boundary: single drivers in each measurement; the AN12 driver is an example, not a guitar speaker; 30 W was noted as softer than concert level. Population: bench loudspeakers; outcomes: harmonic distortion against frequency, fundamental compression against drive, voice-coil resistance against time; evidence class: INTERVENTION. Room, microphone and playback level after the speaker belong to the room the volume and the record vector.

F = Bl(x) i with Bl falling as the coil leaves the gap and Kms(x) stiffening, both displacement-domain and strongest where excursion is largest (below and at resonance); R_e(T) rising with coil temperature cuts current under voltage drive, a gain change with a thermal time constant; above resonance displacement falls as 1/f^2, so the mid and high band is dominated by linear modal filtering.

• S6Yeh, Bank, Karjalainen, Nonlinear modeling of a guitar loudspeaker cabinet, DAFx-08 (2008)Guitar loudspeaker at 30 W, 1.5 m on axis: THD -9 dB at 70 Hz, -25 dB at 100 Hz, -36 dB at 140 Hz, -39 dB at 200 and 400 Hz, -37 dB at 4 kHz; roughly 10 to 30 percent near resonance, about 1 percent at high frequencies; much of the salient timbre comes from linear filtering (INTERVENTION + MODEL).
• S5Klippel Application Note AN12, Causes for Amplitude Compression (undated)Example driver swept 1 to 8 V rms: fundamental reduced 7 dB below resonance, 5 dB at resonance, about 2 dB above it against a linear system; current-based measure isolates thermal from nonlinear compression (INTERVENTION).
• S7Klippel, Perceptual and Physical Evaluation of Guitar Speakers (undated)Guitar loudspeaker at 20 W average and 60 V peak: excursion +4 mm against -3 mm (dynamic DC displacement), voice-coil resistance varying more than 15 percent, 1 to 2 dB amplitude variation within seconds from heating (INTERVENTION).
• S19Klippel, Loudspeaker Nonlinearities: Causes, Parameters, Symptoms (undated)Dominant large-signal nonlinearities are Bl(x), Kms(x), Le(x) and Le(i); low-frequency distortion comes from displacement-driven stiffness and force-factor nonlinearity; break-up irregularities are linear modal behaviour with cone nonlinearity appearing after break-up (MODEL with measurements).

In a comparison of an analog tube chain against a digital model, the measured and perceived difference is governed by the least-matched block inside the declared device-under-test boundary, not by the analog or digital label of the gain stages. For trained listeners judging recorded output, the transducer and its acoustic environment dominate whenever they are not physically shared: the loudspeaker is the largest linear filter in the chain and a major nonlinear generator whose distortion is negligible at small signal and rises steeply with amplitude. A comparison in which one arm drives a real cabinet in a room and the other a headphone or full-range monitor is a comparison of transducers and carries no information about the amplifier model. Where the cabinet is the modeled block, cabinet type sets the result before model architecture does: at 112 dB(A) and one metre, a closed 4x12 modeled by a linear FIR alone reached a 30.2 dB signal-to-error ratio and 20 listeners could not tell it from the recording (45.5 percent correct over 200 ABX responses, 10 trials each), while an open 1x12 under the same protocol was told apart at 78 percent correct (p < .001) at 11.7 dB SER (DAGA 2026; INTERVENTION on 20 listeners, MODEL for the SER). Serious amplifier-model evaluations fix the block: one common cabinet impulse response, one common EQ and loudness matching across all stimuli (Cassidy and De Sena 2023).

The nonlinear loudspeaker parameters Bl(x), Cms(x) and Le(x) vary with cone excursion x, so harmonic and intermodulation products grow faster than linearly with drive; an open-back 1x12 runs at higher excursion for a given SPL than a closed 4x12, so its nonlinear residue is larger and a linear model misses more of it. A microphone at a fixed position samples one point of a radiation pattern set by cone breakup and baffle geometry; two cabinets, or one cabinet at two microphone positions, differ by more decibels across 1 to 5 kHz than any published amplifier model differs from its target.

• S13Klippel, Separated loudspeaker distortion (measurement method page)Force factor Bl(x) and compliance Cms(x) dominate loudspeaker nonlinear distortion; nonlinear distortion is negligible at small signal and rises rapidly with amplitude and depends on stimulus spectrum.
• S4Hybrid Modeling of Guitar Cabinets Using FIR Filters and Neural Networks (DAGA 2026)20 listeners, 10 ABX trials per condition, recordings level-matched to 112 dB(A) at one metre; Table 3: 1x12 linear 78 percent correct at SER 11.7 dB, 1x12 DNN 57.5 percent at 19.8 dB (p = .020), 4x12 linear 45.5 percent at 30.2 dB, 4x12 DNN 50.5 percent at 42.0 dB, neither 4x12 condition above chance.
• S1Cassidy and De Sena, Perceptual Evaluation and Genre-specific Training of DNN models of a High-gain Guitar Amplifier (DAFx 2023)All six neural models of a Peavey 6505 identified above the 95 percent binomial criterion by 16 Tonmeister students over 72 ABX trials each (192 pooled responses per model); the rock-trained LSTM below the 99 percent level; no model distinguishable on the high-gain high-pitched excerpt H, only the rock LSTM indistinguishable on the lowest-gain excerpt F; preference for all seven stimuli inside the indifferent band in a mix; common cabinet IR and loudness matching as the control.
• S14Pakarinen and Yeh, A Review of Digital Techniques for Modeling Vacuum-Tube Guitar Amplifiers (Computer Music Journal)The modeled blocks of a tube amplifier include coupling capacitors, separate amplification stages, the output transformer and the acoustic feedback path from speaker to string; aliasing is inherent in sampled nonlinear modeling; the amplifier model is separated from the cabinet model.

For trained listeners judging recorded guitar through a matched output path, the error-to-signal ratio, null depth or signal-to-error ratio of a model orders models by waveform fidelity but does not predict whether a listener can tell the model from the target. The governing quantity is the share of residual that escapes the guitar signal's masking: inharmonic components, components below the fundamental, and residual in attacks and decays. Measured: in the DAGA 2026 ABX (20 listeners, 10 trials per condition) a neural stage raised the 4x12 cabinet model's SER from 30.2 to 42.0 dB and discrimination did not move (45.5 to 50.5 percent correct over 200 responses each, neither above chance), while on the 1x12 an SER rise from 11.7 to 19.8 dB moved discrimination from 78 to 57.5 percent, still above chance (p = .020). Wright et al. (2020) reported test-set ESR of 0.20 and 0.32 percent (about minus 27 and minus 25 dB) yet 14 critical listeners identified the HT-5 Metal models as different from the reference while many could not hear the 5:50 Plus difference (CORRELATIONAL listening on a MODEL). Cassidy and De Sena (16 trained listeners, INTERVENTION) found the same architecture indistinguishable on the high-gain high-pitched excerpt and distinguishable on the lowest-gain excerpt. Changing the perceptual weighting used in training changed rated quality at constant runtime computation (Wright and Välimäki 2019), and the field's review states a single objective metric cannot replace listening tests.

A distorted guitar signal is dense and strongly harmonic, so it masks error locked to its harmonics; the same signal leaves valleys below the fundamental and between harmonics where small error is exposed, and masking is weaker below a masker than above it (sawtooth aliasing thresholds: 3 to 41 dB of attenuation needed above the fundamental, 0 to 41 dB below). Higher gain raises total residual and raises self-masking at once; the two pull in opposite directions, which is why the high-gain excerpt was easiest to model perceptually and the sparse low-gain excerpt hardest. An energy metric sums error wherever it falls; the ear weights error by where it falls relative to the masker.

• S4Hybrid Modeling of Guitar Cabinets Using FIR Filters and Neural Networks (DAGA 2026)20 listeners, 10 ABX trials per condition, recordings level-matched to 112 dB(A) at one metre; Table 3: 1x12 linear 78 percent correct at SER 11.7 dB, 1x12 DNN 57.5 percent at 19.8 dB (p = .020), 4x12 linear 45.5 percent at 30.2 dB, 4x12 DNN 50.5 percent at 42.0 dB, neither 4x12 condition above chance.
• S3Wright, Damskägg, Juvela and Välimäki, Real-Time Guitar Amplifier Emulation with Deep Learning (Applied Sciences, 2020)Test-set ESR of 0.20 and 0.32 percent; HT-5 Metal models identified as different from the reference while many of 14 critical listeners could not hear the 5:50 Plus difference; aliased components about 40 dB below the first harmonic with no clear audibility on guitar and bass; three minutes of training audio sufficient.
• S1Cassidy and De Sena, Perceptual Evaluation and Genre-specific Training of DNN models of a High-gain Guitar Amplifier (DAFx 2023)All six neural models of a Peavey 6505 identified above the 95 percent binomial criterion by 16 Tonmeister students over 72 ABX trials each (192 pooled responses per model); the rock-trained LSTM below the 99 percent level; no model distinguishable on the high-gain high-pitched excerpt H, only the rock LSTM indistinguishable on the lowest-gain excerpt F; preference for all seven stimuli inside the indifferent band in a mix; common cabinet IR and loudness matching as the control.
• S23Wright and Välimäki, Perceptual Loss Function for Neural Modelling of Audio Systems (2019)Changing the perceptual weighting used during training changed listening-test quality without changing the model's runtime computation.
• S26Vanhatalo et al., A Review of Neural Network-Based Emulation of Guitar Amplifiers (Applied Sciences, 2022)A single objective metric cannot replace subjective listening tests; models with varying time-domain metrics were all judged acceptable in listening; the WaveNet3 HT5M model remained distinguishable from the original. Opened by the anthropic leg; this writer's fetch returned 403.
• S7Lehtonen, Pekonen and Välimäki, Audibility of aliasing distortion in sawtooth signalsForced-choice thresholds: aliased components above the fundamental needed roughly 3 to 41 dB of attenuation for inaudibility, below the fundamental 0 to 41 dB depending on frequency; masking more prominent above the fundamental than below it. Opened by the anthropic leg; this writer's fetch returned 403.

For trained listeners judging level-matched, cabinet-matched recordings through headphones, published discrimination of a state-of-the-art model from its tube target is small and above chance, neither zero nor large. Cassidy and De Sena (2023; 16 Tonmeister students in the ABX, 72 trials each, 12 per model, 192 pooled responses per model; INTERVENTION) found every one of six neural models of a Peavey 6505 above the 95 percent critical level, the rock-trained LSTM (about 57 percent correct by the openai leg's reading of Figure 8) below the 99 percent level. Düvel et al. (2020; N = 177 online listeners passed through a headphone-or-loudspeaker screening, 12 stimuli from a 1968 Marshall rig and its Kemper profile, 1,062 pair responses on an 8-point source-plus-confidence scale; CORRELATIONAL) found d' = 0.34 (range 0.0 to 0.74 across stimuli), 547 of 1,062 responses correct (51.5 percent), and a weak relation to musical sophistication. Wright et al. (2020; 14 critical listeners) found the HT-5 Metal models identifiable and the 5:50 Plus models not for many listeners. On preference, Cassidy and De Sena's 21 listeners rated the real amplifier and all six models inside the minus 10 to plus 10 'indifferent' band in a bass-and-drums mix, and a rock-trained LSTM was rated significantly more realistic than the real amplifier on the lowest-gain excerpt. The population is trained listeners and the task is recorded listening; the result does not transfer to a player in the loop without a separate test. The largest 2026 dataset (NAM A2, 1,184 participants, 105,842 MUSHRA ratings over 37 tones) is rated similarity and carries no identification count.

Discrimination is bounded above by the total variation distance between the two output distributions (ideal observer: P correct = (1 + TV)/2) and a human falls below that bound; a modern model's residual is a few tenths of a percent of signal energy, mostly harmonically locked and masked, so the information available to a listener is small and the measured rate sits a few points above chance. Preference is a different estimand that depends on what a listener likes rather than on what differs, so the same data can show a detectable difference and no preference.

• S1Cassidy and De Sena, Perceptual Evaluation and Genre-specific Training of DNN models of a High-gain Guitar Amplifier (DAFx 2023)All six neural models of a Peavey 6505 identified above the 95 percent binomial criterion by 16 Tonmeister students over 72 ABX trials each (192 pooled responses per model); the rock-trained LSTM below the 99 percent level; no model distinguishable on the high-gain high-pitched excerpt H, only the rock LSTM indistinguishable on the lowest-gain excerpt F; preference for all seven stimuli inside the indifferent band in a mix; common cabinet IR and loudness matching as the control.
• S2Düvel, Kopiez, Wolf and Weihe, Confusingly Similar (Music and Science, 2020)Online discrimination study, N = 177, 12 stimuli from a 1968 Marshall rig and its Kemper profile, 1,062 pair responses, hits 0.526 and false alarms 0.403 (547 of 1,062 correct, 51.5 percent), d' = 0.34, range 0.0 to 0.74, weak relation to musical sophistication.
• S3Wright, Damskägg, Juvela and Välimäki, Real-Time Guitar Amplifier Emulation with Deep Learning (Applied Sciences, 2020)Test-set ESR of 0.20 and 0.32 percent; HT-5 Metal models identified as different from the reference while many of 14 critical listeners could not hear the 5:50 Plus difference; aliased components about 40 dB below the first harmonic with no clear audibility on guitar and bass; three minutes of training audio sufficient.
• S19NAM A2 MUSHRA raw data repository (tone-3000, 2026)105,842 similarity ratings from 1,184 participants over 37 tones, collected April 18 to May 29, 2026, MUSHRA-style with hidden reference and anchor; rated similarity, not identification trials.

Because the best published identification rates sit near 57 to 59 percent correct, a two-alternative forced-choice test needs on the order of 150 pooled trials per condition to detect that rate at 80 percent power and one-sided alpha 0.05; about 60 trials for 65 percent, about 33 for 70 percent, about 620 for 55 percent (binomial power calculation by the anthropic leg and this writer; MODEL class, not a published number). A 10-of-20 result bounds the true rate only to a Wilson 95 percent interval of about 30 to 70 percent; 100 of 200 bounds it to about 43 to 57 percent. Trials from one listener on one excerpt are not independent population observations, so participant-level analysis is required. The informal protocol of asking 'do you hear a difference' carries a response bias independent of the stimuli: in a published critique of an amplifier listening test, listeners reported a difference 61 to 66 percent of the time regardless of whether the amplifiers differed, and an unbalanced same-different mix converts that bias into apparent correctness (ARGUMENT class). An equivalence claim must be stated as a bound (for example, the upper 95 percent limit on identification is below 60 percent under these conditions), never as a failure to reject chance; preference is a separate dependent variable with its own interval.

The standard error of a proportion near 0.5 is 0.5/sqrt(n); detecting a 7-point departure from chance requires that error to be about a third of the departure, hence hundreds of trials. A yes-biased difference count is a criterion shift in signal-detection terms, not sensitivity; d' with a reported criterion separates them. BS.1116, the double-blind triple-stimulus hidden-reference method, is the standard for impairments small enough to need this control, and the EBU recommends it over MUSHRA once impairments are small.

• S1Cassidy and De Sena, Perceptual Evaluation and Genre-specific Training of DNN models of a High-gain Guitar Amplifier (DAFx 2023)All six neural models of a Peavey 6505 identified above the 95 percent binomial criterion by 16 Tonmeister students over 72 ABX trials each (192 pooled responses per model); the rock-trained LSTM below the 99 percent level; no model distinguishable on the high-gain high-pitched excerpt H, only the rock LSTM indistinguishable on the lowest-gain excerpt F; preference for all seven stimuli inside the indifferent band in a mix; common cabinet IR and loudness matching as the control.
• S2Düvel, Kopiez, Wolf and Weihe, Confusingly Similar (Music and Science, 2020)Online discrimination study, N = 177, 12 stimuli from a 1968 Marshall rig and its Kemper profile, 1,062 pair responses, hits 0.526 and false alarms 0.403 (547 of 1,062 correct, 51.5 percent), d' = 0.34, range 0.0 to 0.74, weak relation to musical sophistication.
• S18Stereophile, Blind Listening letters, part 4 (critique of an amplifier listening test)Participants reported a difference 61 to 66 percent of the time regardless of whether the amplifiers differed; an unbalanced same-different mix converts that bias into apparent correctness; a separate test's 52.3 percent correct did not differ from the 50.9 percent chance level. Opened by the anthropic leg; this writer's fetch returned 403.
• S24EBU Tech 3339, Evaluations of multichannel audio codecs (BS.1116 versus MUSHRA guidance)BS.1116 double-blind triple-stimulus hidden-reference is recommended for small impairments, MUSHRA for intermediate ones; listeners who downgrade the hidden reference are excluded.

Every other model error (gain, equalization, nonlinear shape, noise, sag) also exists in analog form and can in principle be matched; foldover of harmonics past the Nyquist frequency exists only in a sampled nonlinear process. Folded components are inharmonic and descend as the note ascends, which defeats the masking the guitar's own harmonics provide. Measured thresholds on sawtooth stimuli in forced-choice listening: aliased components above the fundamental needed roughly 3 to 41 dB of attenuation to be inaudible, those below the fundamental 0 to 41 dB depending on frequency, low-frequency audibility limited only by the hearing threshold (INTERVENTION on trained listeners). In Wright et al. (2020) a neural amplifier model's aliased components sat about 40 dB below the first harmonic at low and mid frequencies and the authors heard no clear aliasing on guitar and bass, an observation and not a threshold experiment. The 2025 engineering state: Esqueda and Murai's antialiased recurrent model shows most aliasing components below about minus 60 dB with no perceivable aliasing in informal listening up to fundamentals of at least 5 kHz; Sato and Smith trade waveform accuracy against alias suppression and leave the listening test to future work (MODEL class, measured spectra). Antiderivative antialiasing reduces low-frequency aliasing by up to about 50 dB in a four-point version. A cabinet filter applied after folding cannot remove an alias already inside the band.

A nonlinearity applied to a band-limited signal generates harmonics at h times f0; in discrete time any component above fs/2 reflects to |h f0 minus k fs| inside the Nyquist interval, where it is no longer a multiple of f0. Because masking is weaker below a masker than above it, and because a rising note drives aliases downward into the sparse region under the fundamental, the audibility specification is a frequency-dependent noise-to-mask ratio, not a THD figure. Oversampling raises fs/2 so fewer harmonics fold; ADAA smooths the nonlinearity's effective bandwidth; both cost computation that real-time hardware must pay.

• S7Lehtonen, Pekonen and Välimäki, Audibility of aliasing distortion in sawtooth signalsForced-choice thresholds: aliased components above the fundamental needed roughly 3 to 41 dB of attenuation for inaudibility, below the fundamental 0 to 41 dB depending on frequency; masking more prominent above the fundamental than below it. Opened by the anthropic leg; this writer's fetch returned 403.
• S3Wright, Damskägg, Juvela and Välimäki, Real-Time Guitar Amplifier Emulation with Deep Learning (Applied Sciences, 2020)Test-set ESR of 0.20 and 0.32 percent; HT-5 Metal models identified as different from the reference while many of 14 critical listeners could not hear the 5:50 Plus difference; aliased components about 40 dB below the first harmonic with no clear audibility on guitar and bass; three minutes of training audio sufficient.
• S6Esqueda and Murai, Antialiased Black-Box Modeling of Audio Distortion Circuits Using Real Linear Recurrent Units (DAFx 2025)Antialiased model spectrum with most aliasing components below approximately minus 60 dB; informal listening with a sine sweep found no perceivable aliasing up to fundamentals of at least 5 kHz.
• S5Sato and Smith, Aliasing Reduction in Neural Amp Modeling by Smoothing Activations (DAFx 2025)A measured trade-off between waveform accuracy and alias suppression in neural amp models; perceptual validation left to future listening tests.
• S25Parker, Zavalishin and Le Bivic, Antiderivative Antialiasing for Memoryless NonlinearitiesThe antiderivative method attenuates low-frequency aliased components, up to about 50 dB in a four-point version, at low oversampling factors. Opened by the anthropic leg.
• S22Shannon, Communication in the Presence of Noise (1949)Band-limited sampling and ideal reconstruction; the distinction between predictable distortion and unpredictable noise.
• S14Pakarinen and Yeh, A Review of Digital Techniques for Modeling Vacuum-Tube Guitar Amplifiers (Computer Music Journal)The modeled blocks of a tube amplifier include coupling capacitors, separate amplification stages, the output transformer and the acoustic feedback path from speaker to string; aliasing is inherent in sampled nonlinear modeling; the amplifier model is separated from the cabinet model.

A digital chain adds conversion, buffering and filter group delay that an analog chain does not, and this delay reaches the player through a mechanism independent of timbre, so no null test removes it and no interactive comparison that leaves loop delay unequal is a timbre test. The measured record, none of it on guitar: with a percussive digital instrument under sub-millisecond latency control, 10 ms with ±3 ms jitter and 20 ms were rated worse on six quality measures than zero added latency while stable 10 ms was not, and metronome synchronisation error did not change (Jack, Stockman and McPherson 2016; replicated in 2018 with 11 amateur musicians and 10 professional percussionists over 12 randomized pairs, professionals more aware of the change; INTERVENTION, within-person). Conscious detection thresholds are far higher: a PEST staircase study (n = 37) found a mean JND of 49 ms at zero base latency, 27 ms at 64 ms base and 77 ms at 512 ms, lower for high musical sophistication (INTERVENTION). Acceptable monitoring latency for performers ranged from 42 ms to possibly under 1.4 ms by instrument and by wedge against in-ear monitoring (Lester and Boley 2007; CORRELATIONAL). On a keypress-to-sound task musicians' delay thresholds averaged about 102 ms against about 180 ms for non-musicians (van Vugt and Tillmann 2014; CORRELATIONAL). The gemini leg's mechanism for sub-5 ms detection, comb filtering between bone-conducted string sound and a delayed monitor, is a hypothesis with no opened measurement; the anthropic leg's prediction that a guitarist's threshold sits nearer 10 to 20 ms because the acoustic string sound is a zero-delay reference is likewise a stated hypothesis.

A causal real-time system cannot emit a response to a pick attack before the attack has been sampled; total delay sums conversion, block buffering, processing and the acoustic path (about 2.9 ms per metre of air). A listener evaluating a recording can have a constant offset removed; a player cannot subtract delay from an ongoing sensorimotor loop, and the quality rating falls at a delay the same player neither consciously detects nor shows in timing error, which is why the two dependent variables must be collected separately. The threshold moves with the operating point (27 ms at a 64 ms base against 49 ms at zero), so a figure at one base does not transfer to another.

• S8Jack, Stockman and McPherson, Effect of latency on performer interaction and subjective quality assessment of a digital musical instrument (Audio Mostly 2016)With sub-millisecond latency control on a percussive digital instrument, 10 ms with jitter and 20 ms were rated worse on six quality measures while stable 10 ms was not, with no change in metronome synchronisation error; the acceptable threshold placed between 10 and 20 ms.
• S9Jack, Mehrabi, Stockman and McPherson, Action-sound Latency and the Perceived Quality of Digital Musical Instruments (Music Perception, 2018)11 amateur musicians and 10 professional percussionists, 12 randomized pairs, conditions 0, 10, 20 ms and 10 ms with ±3 ms jitter; 20 ms and the jitter condition rated worse, 10 ms not different from zero, professionals more aware of latency. Opened by the openai leg; this writer's fetch returned 403.
• S10Schmid, Ambros, Bogon and Wimmer, Measuring the Just Noticeable Difference for Audio Latency (2024)PEST staircase, n = 37: mean JND 49 ms at zero base latency, 27 ms at 64 ms base, 77 ms at 512 ms; lower JND for high musical sophistication.
• S11Lester and Boley, The Effects of Latency on Live Sound Monitoring (AES 123rd Convention, 2007)Acceptable monitoring latency ranged from 42 ms to possibly under 1.4 ms depending on instrument and on wedge versus in-ear monitoring.
• S12van Vugt and Tillmann, Thresholds of Auditory-Motor Coupling Measured with a Simple Task in Musicians and Non-Musicians (PLOS One, 2014)Keypress-to-sound delay detection thresholds: musicians about 102 ms, non-musicians about 180 ms.

An all-analog chain is continuous-time processing without numerical conversion; it is not one galvanically continuous circuit, since coupling capacitors pass AC and block DC between stages, the output transformer transfers energy between separate windings, and every active stage draws its output energy from a power supply, so no current and no electrons travel from pickup to speaker. The two consequences of bidirectional termination that do reach the ear are measurable and large. At the input, pickup inductance and total shunt capacitance form a second-order low-pass with a resonant peak at fr = 1/(2π√(LC)) whose height is set by the load resistance: a Strat-type pickup loaded by an extra 2,200 pF peaks near 2.4 kHz and the same pickup with 47 pF near 8 kHz, and a lower load impedance lowers the peak (MODEL plus measurement). At the output, the power stage's source impedance interacts with the speaker's non-flat Z(f) so that a high source impedance equalizes the driver along its impedance curve; pentode stages have the highest output impedance, triode-strapped the lowest, and less global negative feedback raises it. The output transformer itself adds distortion only below about 100 Hz for a measured Fender unit and 30 Hz for a Hammond (MODEL fitted to measurement). Any buffered analog pedal, active pickup or FET input stage terminates the pickup's view at that buffer, so a buffered analog pedalboard is a cascade of one-way blocks exactly as a digital chain is; a digital chain whose input presents the same complex load and whose power stage or model reproduces the output interaction matches both parameters, while one with a flat 1 megohm converter input and a high-damping full-range power amplifier matches neither and is discriminable for that reason alone.

A passive magnetic pickup is a voltage source in series with an inductance of several henries and a resistance of several kilohms, shunted by cable and input capacitance and by the input resistance; the circuit is a damped resonator whose peak frequency and height are set by the downstream load, which is what bidirectional means here. A tube output stage without heavy feedback has a source impedance comparable to the speaker's nominal impedance, so the speaker's impedance peak at resonance and its inductive rise at high frequency receive more voltage there, an equalization that follows Z(f). Both effects are parameters of the terminations, not of the processing domain; a model that measures and reproduces them removes them from the residual.

• S15USPTO patent 7548850 (pickup and cable loading)Pickup inductance and cable capacitance resonate at fr = 1/(2π√(LC)); a Strat-type pickup with an extra 2,200 pF peaks near 2.4 kHz and with 47 pF near 8 kHz; lower cable capacitance lowers the peak amplitude; load resistance sets the Q.
• S16Damping factor (reference article)Amplifier output resistance varies with frequency in tube and solid-state amplifiers; the frequency-dependent loudspeaker response change caused by output resistance exceeds the damping effect itself; pentode stages have the highest output impedance, triode-strapped the lowest, less global feedback raises it.
• S17Paiva, Pakarinen and Välimäki, Real-Time Audio Transformer Emulation for Virtual Tube Amplifiers (EURASIP JASP, 2011)Measured Fender and Hammond output transformers introduce distortion only below about 100 Hz and 30 Hz respectively; the model is bidirectional, feeding speaker current back to the power amplifier.
• S14Pakarinen and Yeh, A Review of Digital Techniques for Modeling Vacuum-Tube Guitar Amplifiers (Computer Music Journal)The modeled blocks of a tube amplifier include coupling capacitors, separate amplification stages, the output transformer and the acoustic feedback path from speaker to string; aliasing is inherent in sampled nonlinear modeling; the amplifier model is separated from the cabinet model.
• S21Aiken Amplification, Designing a Reactive Speaker Load EmulatorThe LRC network that reproduces a guitar cabinet's impedance curve for an amplifier not driving a speaker; a resistive dummy load has no motional impedance and omits the amplifier-speaker interaction.

As of the legs' searches in October 2026, no published study puts a guitarist blind, in the loop, on a tube chain and its model under shared cabinet, room, level and delay with a discrimination statistic; the nearest record is Buffa and Lebrun (2017), blind comparisons and interviews with four guitarists among several amplifier simulations, qualitative, with configuration latencies from under 1 ms on dedicated hardware to much longer on computer paths. The gap is a gap, not a null. A player closes loops a recording does not: acoustic regeneration from speaker to strings (sustain and controlled feedback, set by SPL, distance and cabinet radiation), the input load loop, the output load loop (source impedance against a reactive speaker; a capture made through a resistive dummy load has no motional impedance and omits the amplifier-speaker interaction, which is why reactive load boxes exist), and thermal and supply nonstationarity over a take. An interactive comparison discriminates on whichever of these the arms do not share before any model error is reached; a tube amp into its own cabinet at 100 dB against a model into headphones differs on three loops and says nothing about the model. An unblinded report ('it breathes', 'it blooms', 'it feels disconnected') is a claim about a mechanism (compression time constant, supply sag, speaker excursion at that SPL, pickup resonance Q, loop delay), and its lawful output is the variable to measure, never a discriminability result. The largest 2026 dataset (NAM A2 MUSHRA) is rated similarity of recordings and does not fill the gap.

With acoustic feedback the chain is y = G(x_hand + H y), so y = G/(1 minus G H) x_hand; near a strong feedback condition small differences in gain, phase or delay between arms produce large differences in closed-loop behavior, and matching a prerecorded performance (open loop) is weaker than matching the response to a family of player actions (closed loop). The loudspeaker's motional impedance returns a voltage to the output stage that a resistor does not, so the amplifier's distortion and frequency response under a real speaker differ from those under a dummy load; a model captured through a resistor models a different operating point.

• S20Buffa and Lebrun, Real time tube guitar amplifier simulation using WebAudio (2017)Blind comparisons and interviews with four guitarists among several amplifier simulations, qualitative; measured latencies from under 1 ms on dedicated modeling hardware to much longer on computer paths.
• S21Aiken Amplification, Designing a Reactive Speaker Load EmulatorThe LRC network that reproduces a guitar cabinet's impedance curve for an amplifier not driving a speaker; a resistive dummy load has no motional impedance and omits the amplifier-speaker interaction.
• S13Klippel, Separated loudspeaker distortion (measurement method page)Force factor Bl(x) and compliance Cms(x) dominate loudspeaker nonlinear distortion; nonlinear distortion is negligible at small signal and rises rapidly with amplitude and depends on stimulus spectrum.
• S14Pakarinen and Yeh, A Review of Digital Techniques for Modeling Vacuum-Tube Guitar Amplifiers (Computer Music Journal)The modeled blocks of a tube amplifier include coupling capacitors, separate amplification stages, the output transformer and the acoustic feedback path from speaker to string; aliasing is inherent in sampled nonlinear modeling; the amplifier model is separated from the cabinet model.
• S19NAM A2 MUSHRA raw data repository (tone-3000, 2026)105,842 similarity ratings from 1,184 participants over 37 tones, collected April 18 to May 29, 2026, MUSHRA-style with hidden reference and anchor; rated similarity, not identification trials.